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 (Insert > 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)

128 comments

  1. Hi, If there's a code say 3000111018609 and a calculation of even and odd digits in the code, 3+0+9+3(1+8+0)=TOTAL. If the digit 8 is missing in the code, how can I create a formula which finds the missing digit especially those inside the brackets? Thanks

    1. Hello Jim!
      Sorry, I have no idea exactly what the task is. I don't see relationship between the code and this formula. And this formula is mathematically incorrect and cannot be calculated. Please clarify your specific problem or provide additional information to understand what you need.

  2. Hello,

    Using the FindSumCombinations macro/formula the summation formula doesn't seem to be able to solve past 1500. What part of the code do I need to change for the solver to produce this?

    Kind Regards,
    Gary

    1. Hello,

      An add on to this problem. If I had more variable numbers to find the exact number it generates past 1500 but not for 2000. I have 8 variable numbers (50,100,150,200,250,300,400,450), not having 450 in the set gives me a #value error. Which part of the code will have to change in order for the macro to work with less variability?

      I'm very new to macros and spending my time reviewing the code is proven difficult.

      1. Hi! Sum of the numbers 50,100,150,150,200,250,300,400 is 1450. If you want the sum of 1500, you will get an error.

  3. Does the macro work with negative numbers ?

    1. Hello Aaron!
      You can download sample file linked at the end of the article. Use negative values in this file and you will see that the macro works with them.

  4. I'm using the Macro and works brilliantly, but I need it to work with currency for example a list of values like £4,562.32

    It only seems to work with whole numbers and not numbers with decimal places

    Basically I have a list of invoice values that i am trying to match with a value that was debited from our buisness account

    Any help much appreciated

    1. Hello Paul!
      As I wrote in the comments below, multiply all your numbers by 100 and use the macro. To get numbers with two decimal places, divide the results by 100.

  5. for the userform section - to find the refedit box:

    Tools > Additional Controls

    Scroll down until you find it, click on it.

    Hit ok

  6. Just if each output string was written one below the other in a column, will be better and It would be much easier to use, especially when there are a lot of states obtained.

    1. I see now. You explained it before. Thanks a lot again.

  7. Great. Thank you.

  8. Hi Alex,
    This is so interesting. I've been trying to find a formula to get the combination of boxes that has X qty and evenly distribute them depending on how many group. Can you help on how to get the box id combination so I can get the maximum qty combination.

    For example below:
    18694 (the maximum combination that is less than 19213 using manual calculation)

    LotID LOT QTY # OF BOX BOXID BOX QTY # OF GROUP QTY PER SECTION (LOT QTY / 2) BOX COMBINATION
    A 38427 21 2 1726 2 19213 6,7,10,14,16,17,18,20,22,24
    A 38427 21 3 1804
    A 38427 21 4 1806
    A 38427 21 5 1776
    A 38427 21 6 1874
    A 38427 21 7 1855
    A 38427 21 8 1798
    A 38427 21 9 1836
    A 38427 21 10 1855
    A 38427 21 11 1814
    A 38427 21 12 1851
    A 38427 21 13 1837
    A 38427 21 14 1865
    A 38427 21 15 1692
    A 38427 21 16 1858
    A 38427 21 17 1854
    A 38427 21 18 1865
    A 38427 21 20 1863
    A 38427 21 21 1793
    A 38427 21 22 1937
    A 38427 21 24 1868

  9. Hi,

    I have a set of dealers with their SKU wise sale in qty, I want to find out the top 50% / 30%/ 10% Sku sales for each dealers

    Can you help me in this please.

    Regards
    vinod

    1. Hi! Assume you have the following columns in your dataset: Dealer, SKU, Sales Quantity.
      Sort the Data: First, sort by Dealer.
      Then, sort by Sales Quantity in descending order. Read more: How to sort in Excel by row, column names and in custom order.
      Add a column D called Cumulative Sum.
      Add another column E called Cumulative Percentage.
      For example, let's assume your data starts from row 2 (with headers in row 1):
      In cell D2 (with headers in row 1), use the SUMIF formula:
      =SUMIF(A$2:A2, A2, C$2:C2)

      In cell E2, use the formula:
      =D2/SUMIF(A:A, A2, C:C)

      Copy it down along the column.
      Create new columns for Top 50%, Top 30%, and Top 10%.
      In cell F2 (Top 50%), use the IF formula:

      =IF(E2<=0.5, "Top 50%", "")

      This will help you identify the top 50% SKU sales for each dealer.

  10. hi,
    thanks for a great solution...

    why is arNumbers(indI) changing 103.99 to 104???

    peter

    1. Hi Peter!
      The Long data type in VBA can only store whole numbers (integers), so indI can't take on a value like 103.99.

  11. Hi,

    This is a great solution and works well - I wondered if you could help me with a query though. instead of returning the number values is it possible to return the values one column to the left (which in my case are text descriptions?)

    Thanks! :)

      1. Hi,

        Thanks for the response - I've been using the custom function and my dataset has some duplicate numbers (exactly the same) and also they are monetary figures so have a 2 decimal places.. so not sure INDEX MATCH would provide what I need.

        I've noticed that some of the results aren't accurate to 2 DP.. i think there must be some rounding or tolerance in how it provides the results? - not sure how to improve this - a neat solution I just wished it was a little more robust/accurate

        1. Hello Paul!
          To avoid rounding problems in Excel, I recommend using the integer numbers. If your numbers have two decimal places, you can multiply them by 100, do a calculation, and then divide the chosen numbers by 100.

  12. find all combinations that equal a given sum in excel without round

  13. How to find the largest possible whole value instead of a given value. I'm work with large decimal datasets and would like to solve for the largest possible whole number instead of playing around with given values.

      1. Hi! unfortunately I was not able to figure it out with the linked article.

        Please see below example for finding the largest possible integer sum of the dataset

        0.007
        0.0055
        0.0027
        0.0027
        0.0025
        0.0024
        0.0029
        0.0027
        0.0077
        0.0079
        0.115
        0.1426
        0.2245
        0.081
        0.1348
        0.162
        0.7962
        0.2116
        1
        1
        0.532
        0.6756

        From changing the value I enter in the "target sum", I found the max to be 4, however, I was wondering if there is a way to have it solve directly for the combination that will sum to the largest possible integer of the dataset instead of playing around with target sum. This will be really helpful if I have a larger data set with higher values to solve for.

        1. Hi! It is not possible for me to set up all the necessary rules for the Solver tool for you in your workbook. In this blog, we give advice to users, but we don't do their work for them.

  14. Hi! I was wondering if there is a way to find combinations containing repetitions, as this formula given does not allow for repeats.
    For example, I want to find possible combinations to achieve sum of 100, and my options are 10, 20, 30, 90. With this formula, the only combination identified is 10, 90.
    Is there a way to show other combinations like 10, 30, 30, 30 or 20, 20, 30, 30, for example? Thank you!

    1. Hello Penny!
      Unfortunately these functions and macros only allow you to use each value once. If each value can be used multiple times, the number of possible combinations is huge.

  15. The articlue is excelent !!!! Thanks for sharing !!
    The solution works fine.
    I'm interested in getting the list of combinations in a given range, not only de exact value.
    Is there a way to achive this?
    I read a couple of months ago that the solution wasn´t available but I wonder if it is now,

    Regards,

    Marcelo

    1. Hi! To get all combinations in a given range, you need to modify the VBA code of the function presented above. We do not modify VBA code on request, but you can try to do it yourself.

  16. Hi.

    I tried the solver method and I noticed it does not work for over 200 cells. Please what can I do if I'm working with over 1000 cells?

  17. The requirement is to display all the combination from the list of decimal entries for the value of Total sum without repetition of numbers with condition only 6 entries to be consider for SUM.

    For Example: SUM is 50, Table entries are 1- 30 numbers with decimals . Here the condition is for total SUM 50, only any 6 entries should choose without repetition of values from the initial table .
    Output : Display all the combination for the SUM 50 in column

    1. Hi! Try using the function provided in the article above. We do not do VBA code creation or customization on request.

      1. Thank you for prompt response!

        The above solution doesn't meet my requirement.

        Wishing you great day ahead.

  18. I actually want something different.
    I want a formula to cut the values from the range whose sum equals to 228 and paste them in separate row.
    then value being cut down it again pick values from the data range and paste them again in different row.
    and this goes so on until no value remain in data range whose sum equal to 228.

    Please reply

  19. How to find all combinations of numbers that equal given sum in Excel?
    This was my question and the solutions you have given is superb and it worked. I'm a regular Excel user hence this information too important to me.
    Thank you so much.

  20. Hi, this is great. thank you.

    If my list does not contain data that add to my specific number (& therefore, solver is unsuccessful), is there a way that excel can output the "closest" value that can be reached using the data?

    For example if I have list of data & I'm trying to find 4000.
    But it is no possible and solver "unsuccessful".
    But, for example, 3999 is possible and close to my value.

    Thanks,

    1. Hi! To find the "closest" value to 4000 in the target cell A1 that can be reached using the data, you can write the formula =ABS(A1-4000) in cell B1. You can then use the Excel Solver to find the minimum value in cell B1 by modifying cell A1.
      I hope it’ll be helpful.

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