How to find all combinations of numbers that equal given sum in Excel

3 ways to find all combinations from a set of numbers that sum to a specific total.

Finding combinations of values that add up to a given sum is a common problem in data analysis. For example, you may want to know all possible combinations of items that can be purchased with a given budget, or all possible ways to allocate resources to meet certain requirements. In this article, we will explore how to use Excel Solver and VBA to accomplish the task. We will also discuss some potential pitfalls and limitations of each approach that may arise when handling large datasets or complex constraints.

Find a combination of numbers equal to a given sum with Excel Solver

Regrettably, none of inbuilt Excel functions can help you identify the numbers that add up to a given total. Luckily, Excel provides a special add-in for solving linear programming problems. The Solver add-in is included with all versions of Excel but is not enabled by default. If you are not familiar with this tool, here's a great article on how to add and use Solver in Excel.

With the Solver add-in activated in your Excel, proceed with the following steps:

  1. Create the model.

    For this, enter your set of numbers in one column (A3:A12 in our case) and add a blank column to the right of your numbers for the results (B3:B12). In a separate cell (B13), enter a SUMPRODUCT formula similar to this:

    =SUMPRODUCT(A3:A12, B3:B12) Creating a model for Excel Solver

  2. Run the Solver add-in.

    On the Data tab, in the Analysis group, click the Solver button. Run the Excel Solver add-in.

  3. Define the problem for Solver.

    In the Solver Parameters dialog box, configure the objective and variable cells:

    • In the Set Objective box, enter the address of the formula cell (B13).
    • In the To section, select Value Of and type the desired sum value (50 in this example).
    • In the By Changing Variable Cells box, select the range to be populated with the results (B3:B12).
    Define the problem for Solver.
  4. Add the constraints.

    To specify the constraints, i.e. the restrictions or conditions that must be met, click the Add button. In the Add Constraint dialog window, select the resulting range (B3:B12) and pick bin from the drop-down list. The Constraint will be automatically set to binary. When done, click OK. Add the constraints.

  5. Solve the problem.

    When taken back to the Solver Parameter dialog window, review your settings and click the Solve button. Solve the problem.

    A few seconds (or minutes) later, the Solver Results dialog box will appear. If successful, select the Keep Solver Solution option, and click OK to exit the dialog. Keep Solver solution.

As a result, you will have 1 inserted next to the numbers that add up to the specified sum. Not a user-friendly solution, but it's the best that out of the box Excel can do.

For visualization purposes, I've highlighted the cells that give the desired sum in light green: Find a combination of numbers that equals a specified sum.

Limitation: Excel Solver can find, at most, just one combination of numbers that equals a specific sum.

Find all combinations that equal a given sum with custom function

To get all possible combinations from a given set of numbers that add up to a certain value, you can use the custom function below. If you are new to UDFs, you'll find a lot of useful information in this guide: How to create custom user defined functions in Excel.

Custom function to find all combinations that equal a given sum
Option Explicit Public Function FindSumCombinations(rngNumbers As Range, lTargetSum As Long) Dim arNumbers() As Long, part() As Long Dim arRes() As String Dim indI As Long Dim cellCurr As Range ReDim arRes(0) If rngNumbers.Count > 1 Then ReDim arNumbers(rngNumbers.Count - 1) indI = 0 For Each cellCurr In rngNumbers arNumbers(indI) = CLng(cellCurr.Value) indI = indI + 1 Next cellCurr Call SumUpRecursiveCombinations(arNumbers, lTargetSum, part(), arRes()) End If ReDim Preserve arRes(0 To UBound(arRes) - 1) FindSumCombinations = arRes End Function Private Sub SumUpRecursiveCombinations(Numbers() As Long, target As Long, part() As Long, ByRef arRes() As String) Dim s As Long, i As Long, j As Long, num As Long, indRes As Long Dim remaining() As Long, partRec() As Long s = SumArray(part) If s = target Then indRes = UBound(arRes) ReDim Preserve arRes(0 To indRes + 1) arRes(indRes) = ArrayToString(part) End If If s > target Then Exit Sub If (Not Not Numbers) <> 0 Then For i = 0 To UBound(Numbers) Erase remaining() num = Numbers(i) For j = i + 1 To UBound(Numbers) AddToArray remaining, Numbers(j) Next j Erase partRec() CopyArray partRec, part AddToArray partRec, num SumUpRecursiveCombinations remaining, target, partRec, arRes Next i End If End Sub Private Function ArrayToString(x() As Long) As String Dim n As Long, result As String result = x(n) For n = LBound(x) + 1 To UBound(x) result = result & "," & x(n) Next n ArrayToString = result End Function Private Function SumArray(x() As Long) As Long Dim n As Long SumArray = 0 If (Not Not x) <> 0 Then For n = LBound(x) To UBound(x) SumArray = SumArray + x(n) Next n End If End Function Private Sub AddToArray(arr() As Long, x As Long) If (Not Not arr) <> 0 Then ReDim Preserve arr(0 To UBound(arr) + 1) Else ReDim Preserve arr(0 To 0) End If arr(UBound(arr)) = x End Sub Private Sub CopyArray(destination() As Long, source() As Long) Dim n As Long If (Not Not source) <> 0 Then For n = 0 To UBound(source) AddToArray destination, source(n) Next n End If End Sub

How this function works

The main function, FindSumCombinations, calls a few subsidiary functions that implement smaller sub-tasks. The function named SumUpRecursiveCombinations executes the core algorithm that finds all possible sums in the specified range and filters those that reach the target. The ArrayToString function controls the form of output strings. Three more functions (SumArray, AddToArray, and CopyArray) are responsible for processing intermediate arrays: each time we create a temporary array, add one element from the source array to it, and check if the target sum is reached. The core algorithm is taken from this Stackoverflow thread, thank you guys for sharing!

Syntax

From the user's perspective, the syntax of our custom function is as simple as this:

FindSumCombinations(range, sum)

Where:

  • Range is the range of numbers to test.
  • Sum is the target sum.

Note! The custom function only works in Dynamic Array Excel 365 and 2021.

How to use the FindSumCombinations function:

  1. Insert the above code into a Code module of your workbook and save it as a macro-enabled workbook (.xlsm). The step-by-step instructions are here.
  2. In any blank cell, enter a FindSumCombinations formula and press the Enter key. Make sure there are enough empty cells to the right to output all combinations, otherwise the formula will return a #SPILL error.

For example, to find all possible combinations of numbers in the range A6:A15 that equal the sum in A3, the formula is:

=FindSumCombinations(A6:A15, A3)

As with any other dynamic array function, you enter the formula is just one cell (C6 in the image below) and it populates the results into as many cells as necessary. By default, the comma-separated strings are output in a row: Find all combinations of numbers that equal a given sum.

To return the results in a column, wrap the custom function into TRANSPOSE like this:

=TRANSPOSE(FindSumCombinations(A6:A15, A3)) Return combinations of numbers that equal a given sum in a column.

To output the strings in the form of an array enclosed in curly braces, modify the ArrayToString function as follows:

Private Function ArrayToString(x() As Long) As String Dim n As Long, result As String result = "{" & x(n) For n = LBound(x) + 1 To UBound(x) result = result & "," & x(n) Next n result = result & "}" ArrayToString = result End Function

The results will look similar to this: Return combinations that equal a certain sum in the form of an array.

Limitation: This custom function works only in Dynamic Array Excel 365 and Excel 2021.

Get all combinations that equal a given sum with VBA macro

The custom function described above returns the combinations of numbers as strings. If you'd rather have each number in a separate cell, this macro will be helpful. The code is written by another Excel expert Alexander Trifuntov who has been actively helping users to solve various Excel problems on this blog.

Start by adding the following code to your workbook. For the detailed steps, please see How to insert VBA code in Excel.

Macro to find all combinations that add up to a given value
Public RefArray1 As String Public DS As Variant Public TargetSum As Long Public TargetCol As Integer Public TargetRow As Integer Sub Combination() UserForm1.Show End Sub Function GrayCode(Items As Variant) As String Dim CodeVector() As Integer Dim i, kk, rr, col1, row1, n1, e As Integer Dim lower As Integer, upper As Integer Dim SubList As String Dim NewSub As String Dim done As Boolean Dim OddStep As Boolean Dim SSS Dim TargetArray() As String kk = TargetCol rr = TargetRow col1 = TargetCol + 3 row1 = TargetRow OddStep = True lower = LBound(Items) upper = UBound(Items) Cells(rr - 1, kk) = "Result" Cells(rr - 1, kk + 1) = "Sum" Cells(rr, kk + 1) = TargetSum Cells(rr - 1, kk).Font.Bold = True Cells(rr - 1, kk + 1).Font.Bold = True ReDim CodeVector(lower To upper) 'it starts all 0 Do Until done NewSub = "" For i = lower To upper If CodeVector(i) = 1 Then If NewSub = "" Then NewSub = "," & Items(i) SSS = SSS + Items(i) Else NewSub = NewSub & "," & Items(i) SSS = SSS + Items(i) End If End If Next i If NewSub = "" Then NewSub = "{}" 'empty set SubList = SubList & vbCrLf & NewSub If SSS = TargetSum Then Cells(rr, kk).NumberFormat = "@" Cells(rr, kk) = "{ " & Mid(NewSub, 2) & " }" TargetArray() = Split(Mid(NewSub, 2), ",") n1 = UBound(TargetArray) For e = 0 To n1 Cells(row1, col1) = TargetArray(e) row1 = row1 + 1 Next e col1 = col1 + 1 row1 = TargetRow rr = rr + 1 End If SSS = 0 'now update code vector If OddStep Then 'just flip first bit CodeVector(lower) = 1 - CodeVector(lower) Else 'first locate first 1 i = lower Do While CodeVector(i) <> 1 i = i + 1 Loop 'done if i = upper: If i = upper Then done = True Else 'if not done then flip the *next* bit: i = i + 1 CodeVector(i) = 1 - CodeVector(i) End If End If OddStep = Not OddStep 'toggles between even and odd steps Loop GrayCode = SubList End Function

Next, create a UserForm with the following design and properties: Create a UserForm.

After completing the form's design, add the code for the form. For this, right-click on the form in the Project Explorer and select View Code:

Code for the UserForm
Private Sub CommandButton1_Click() Dim B Dim c As Integer Dim d As Integer Dim A() As Variant Dim i As Integer Dim e As Integer DS = Range(RefEdit1) TargetSum = TextBox1.Value Range(RefEdit2).Select TargetCol = Selection.Column TargetRow = Selection.Row c = LBound(DS) d = UBound(DS) ReDim B(d - 1) For i = 1 To d e = i - 1 B(e) = DS(i, 1) Next i Call GrayCode(B) Unload Me End Sub Private Sub Label1_Click() End Sub Private Sub Label3_Click() End Sub

Tip. Instead of re-creating the form from scratch, you can download our sample workbook at the end of this post and copy the Code module and UserForm1 from there (see how to copy a macro to another workbook). To make the macro accessible in all your workbooks, copy it to the Personal Macro Workbook.

With the code and form in place, press Alt + F8 and run the FindAllCombinations macro: Run the FindAllCombinations macro.

On the form that pops up, define the following:

  • The range with the source numbers (A4:A13)
  • The target sum (50)
  • The upper-left cell of the destination range (C4).

When specifying the upper-left cell of the output range, make sure there is at least one blank row above (for headers) and enough blank cells down and to the right. If there are insufficient blank cells, your existing data will be overwritten. Macro to return all combinations that equal a specified sum.

Clicking OK will produce the result like shown in the screenshot below:

  • In C4:C6, you have the combinations of numbers as comma-separated values.
  • Columns F, G and H hold the same combinations of numbers, with each number residing in a separate cell.
  • In D4, you have the target sum.
Result - combinations of numbers with each number residing in a separate cell.

This form of output makes it easier to check the result - just enter the SUM formula in cell F13, drag it to the right across two more cells and you will see that each combination of numbers adds up to the specified value (50). Use the SUM formula to check the result.

Limitation: For a large set of numbers, the macro may take some time to generate all possible combinations.

Advantages: Works in all versions of Excel 2010 - 365; provides two forms of output - strings of comma separated values and numbers in separate cells.

In conclusion, finding all combinations of values that equal a certain value is a powerful tool for solving a wide range of data analysis problems. Hopefully, this article has provided you with a useful starting point for exploring this topic further and handling similar problems in your own work. Thank you for reading!

Practice workbook

Find combinations that equals a given sum - examples (.xlsm file)

Table of contents

108 comments

  1. I want a random combination of my sum but not all the combination one is enough. But I have ample data, so when I use your function my Excel freezes because it looks for all the combinations. How do I stop that?

  2. I wonder if we could allow repetition of numbers in the combination in order to make possible a better approach of the final sum? How would that be possible? For exemple, if we consider a set of numbers 1,2,3,4 and we want to look for combinations that sum up to 20 we must allow he numbers in the combination to be repeatable is this case included in this approach by SOLVE function?

  3. Hi, my computer can't handle the large arrays I want to calculate. I also do not need combinations of >4-5 values to sum my desired value. Is there a way to edit this function to only search for a maximum set of values to sum to 1) match my needs better 2) to lessen the burden on my computer?

    Ex:
    (array: 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) (sum target: 24)
    desired yield: (15,9), (6,7,9), etc
    undesired/wasteful yields: (1,2,3,7,9)

    • Hi! In order to select the part of the values you want, Excel must calculate all values. Therefore, your request is not possible.

  4. Hi, am trying to use the macro to work with dollar amounts (dollars and cents) and am getting a runtime error. Using whole integers works fine if the data entered does is not greater than .00, all other amounts will compile an error. Is there a way to alter the code to include decimals. For instance, search for $1043.02 from the following data.

    $20.67
    $129.61
    $193.26
    $74.34
    $66.61
    $188.67
    $125.96
    $132.79
    $169.81
    $99.73
    $95.80
    $339.79
    $410.21
    $293.02
    $142.90
    $298.89
    $262.48
    $1,611.81
    $384.78

    • Hello! This question has been asked before, if you've read the comments. The function works with integers. You can multiply all your numbers by 100.

      • Yes, indeed I did see that, but I was asking how to alter the code to avoid the workaround itself. When altering the search values and the target value by 100 factor, a run-time error 6 "overflow" results with the debug line pointing to "TargetSum = TextBox1.Value" The search seems to work find for smaller integers, but for the larger numbers error out. Otherwise this is great, because it seems to also take negative values into consideration, which is perfect for the scenario I am trying to use if the larger numbers work.

        • To further explain, if you look at the first 5 values I provided and alter them to

          2067
          12961
          19326
          7434
          6661

          Their sum is 48449. When using this value in the target value box, the mentioned runtime error appears.

            • Thanks for looking into this. I can't seem to get it to work while changing the code, perhaps I am not editing everything correctly. Could you upload a new sample workbook to work off of with the changes you've made? I appreciate your time!

              • Thank you! It seems to run in the background now but is freezing up my excel every time. I've tried with the previous example I gave earlier, as well as a sum of just 2 numbers to see if it was just a calculation/processing issue on my end with no luck - just crashes. I'm assuming the sample workbook works fine for you?

              • Hi! The speed of Excel depends on such things as the formatting of your data, other processes running at the moment, available system resources, RAM, etc. We don't have any problems with this file.

              • I assume general, currency, and number formatting for cells should work the same or is there one cell format we should be sticking to?

  5. Hi! Thank you so much for the great work. Whenever I try to run the macro function (copied from the example file) I receive a "Run-time error 6: Overflow" message. Thanks again!

    • Hi! Unfortunately, I can't see what you're doing or what data you're using. Try placing your data in a sample file and find the combination of numbers.

  6. Can you please change the code so the output are only groups of 5 integers which satisfy the same requirement?
    Thank you in advance.

  7. Hello, I am using the findsumcombinations function and it is rounding to the nearest dollar, when I need it to be exact on the penny.

    Thoughts on this?

    thank you!

  8. hello,

    we have steel coil slitting equipment. we are required to have minimum wastage from our slitting plan. lets keep that at 0-5mm from the coil width. suppose the coil width (which differs) here is 1219mm and considering the minimum wastage we have 1214mm remaining from which we can take as many slits as required from a list of slit sizes.
    e.g 121mm / 240mm / 35.70mm / 115mm / 95.4mm (with decimals)

    now i need a best combination list, multiples of each size allowed, where the total of the selected size should not exceed 1219-5=1214mm

    kindly advice how can this be achieved.

  9. Hello,

    I did a simple Bing search for "Find all combinations of a number..." and it led me to this post. Excellent work, I love the simplicity and easy instructions. I will dabble with the UDF in a bit, but for now, could you tell me how to do the above method and have the following additional information.

    Not only show a 1, but also append the times it needs to be multiplied after adding the other numbers in the array.

    For example

    Desired total would be 50
    Left array is 1 through 10
    The empty column results would show 1 x10 next to the 5
    The 1 indicates true, while the x(number) indicates multiples needed

    Of course, the 10 would be 0 in the right array because 5x10 already reaches the desired total.

    Please let me know if this makes sense.

    • Hi! This custom function and macro do not assume that the numbers will be used more than once. Also, they show all possible combinations of numbers. For your case, it is better to use Solver.

  10. Hi there,
    First I would like to thank you very much for this great work,
    I was looking for this solution for a while since I work in accountancy and I struggle to find the possible solutions or combinations that add up to a given value say payment.
    I have one question to ask if you don't mind,
    in the VBA macro, I don't see anything to change like a range or a sheet name,
    My question is, does it work on any workbook as it is ?
    in other words, would it work if I copied it to my workbook with my tables and values or is there anything to modify ?
    Thanks again for your efforts
    Great job.
    May God bless you.

    • Hi! Copy the code into your workbook as described in the article above. Remember to save it as an .xlsm
      The macro will work with your data.

  11. By the way great job. May God bless you.

  12. Hi. I am using findsumcombinations formula, even there should be more than 1 result, excel gives me only one of them. Is there any solution for that?

      • Just I wrote 1, 2, 3, 4, 5 in each sells and my formula is "=FindSumCombinations". The sum should be 10. The only option excel gives is 1, 2, 3, 4. While we have 1+4+5 for example. It gives only one result.

        • 10 is in cell D10. The numbers 1,2,3,4,5 are in cell D12, D13, D14, D15, D16. And my formula is : =FindSumCombinations(D12:D16;D10)

          And the result is only "1,2,3,4"

        • Look carefully at your formula. I'm sure you only used numbers 1,2,3,4. The formula FindSumCombinations with numbers 1,2,3,4,5 gives 3 results {1,2,3,4} {1,4,5} {2,3,5}

  13. Hi Newbie here, May I know why after I press Alt + F8 and run the FindAllCombinations macro it show Run-time error '424' and ask for Debug. Did I skip a step

    • Hi! I cannot see all of your actions. You may not have followed all of the instructions in the article above. Download the example file from the link at the end of the article.

  14. I am struggling with nimber with decimals. Like money amounts. How can I get it to work with decimals.

      • When I am looking for an amount with a decimal point as 788.39
        And the numbers I'm working with is 45.35, 54.98, 526.35, 54.34, 98.36, 54.36, 59.47, 85.64, 98.63, 78.36 It gives me an error.

        But if i use 45.00, 54.00, 526.00, 54.00, 98.00, 54.00, 59.00, 85.00, 98.00, 78.00 Looking for 788.00 it will work.

        What do I need to format so when the number does have a decimal point it will work.

        Thank you very much for your help.

        • Hi! I can't guess which error you are getting. Also note what decimal separator Windows uses. Also check if some number is written as text.

        • It is designed for integers coz they are using Long data type

  15. Thank you for putting together this great tutorial and example. Out of the many out there, yours is most effective while also being easy to follow. Thank you! Also, I must say that you have the patience of saints to continue answering nearly the same exact questions from people who don't follow the tutorial and therefore cannot apply it to their specific situation. Thanks again for sharing your brilliance with the world!

  16. I need select 6 unique numbers from a range say (1 to 45) that the sum is as specified say 165
    What is the formulae for all possible combinations (I know it is 72,974, but I don't know how to get it)?
    How to generate all the possible 6 number combination.
    Thanks in advance
    Regards
    Steven

    • Hi! You can use the function or macro from the article above, if your Excel has the technical capabilities to perform these calculations.

      • Alexander,
        Thanks for your response, but the spread sheet uses all numbers required to get the sum.
        I need to specify the number of selected numbers say 6, from a pool of 45 to give the required sum, say 165
        Any Idea on the formula, for all possible combination of say 6 unique numbers from a specified range say 1 to 45, to give the specified sum, say 165?

        Regards, Steven

  17. Hi, may I ask if column A is combines data separate with comma and the value is 1,2,3 and column B is another cell with a combined date 7,8,9 . How can I come up with the formula to get the results of the first number show in column A to multiply the first number in column B and so on for example like (1*7)+(2*8)+(3*9)? Thanks.

  18. For a large data sets, Excel goes "Not responding" and I have observed that it uses only 2 thread of my 12 thread CPU, how can we tweak it to support multi threading.

      • Hi Alexander,

        Thank you for your prompt response, even though it is enabled, excel still uses 2 threads.
        Also, how can we make changes to the formula if we wish to find the combinations using a set of specified number. For example, if we have 14, 80, 6, 60, 15, 25, 90, 3, 7 and we want to find all sets of 3 which would add upto 100

          • I somehow managed to solve the set requirement. Would greatly appreciate if you could help with multi-threading challenge

  19. Suppose I have a list of range as follows:

    15
    25
    32
    53
    85
    45
    14
    46
    45
    32
    40
    85

    I wish to get the combination of all possible numbers which on adding gives me the value between 145 to 150 (inclusive both).

    There can be multiple combinations possible which gives me the value between 145 to 150 but find all of them manually is not so easy.

    Is there any way we can get my required result through Excel .

    Please tell me the way.

  20. How to find all combinations of numbers that equals (say, 25000) or falls between a range (say, 22,500-25,000) of given sum in Excel.

    Is it possible to find the sum of numbers that is within a range ?

    If yes please tell me how.

    • To find all combinations of numbers that equal a given value, say 25000, use one of the methods described in this tutorial. For finding a sum that falls within a range, we don’t have a ready-made solution at the moment, sorry. Writing such code will take a while.
      To find the sum of numbers within a range, use the SUM function :)

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