It's possible with the Solver add-in*. The following steps worked for me in Excel 2007 and 2010.

1. Designate a cell to hold the result (C1 for this example) - this is the target cell, and a column that excel can use for scratch-work (B1:B100 for this example)
2. In the target cell, enter the formula "=SUMPRODUCT(A1:A100,B1:B100)" (no quotes). This will calculate the sum of A1*B1+A2*B2+...etc
3. Select Open the solver (Data tab, Analysis group)
4. The target cell should be obvious (1 for this example)
5. For 'Equal To:' select 'Value of:' and enter the desired value
6. In the 'By Changing Cells' enter "1:100" (no quotes, and it may be necessary to initialize these values to 0 yourself)
7. Add a constraint to the cells that can be changed. In the pull-down, select 'bin' (Binary). This restricts the values of these cells to 0 (removing the corresponding A cell from the sum) or 1 (adding the corresponding A cell to the sum).
8. Click 'Solve' and wait. The numbers that are part of the subset you're looking for will have a 1 in the B column

If the solver is taking a long time, you can help it out by removing rows that obviously won't work (total is in dollars, and only one row has nonzero cents)

Bonus: You can have excel automatically highlight the cells that you're looking for by adding conditional formatting to those cells. Select all of the cells you want to format and from (Home tab)>>(Styles group)>>Conditional formatting>>New Rule select 'Use a formula to determine which cells to format'. In the formula, enter '=$B1=1' (no quotes) which will evaluate to true if the corresponding row in the B column is 1. For the format, you can add whatever you want (bold, italic, green fill, etc).

Another easy way to find the important rows is to sort column B Z->A, and all the 1's will come to the top.

*The solver add-in can be installed with these steps

1. Click the Microsoft Office Button, and then click Excel Options.
2. Click Add-Ins, and then in the Manage box, select Excel Add-ins.
3. Click Go.
4. In the Add-Ins available box, select the Solver Add-in check box, and then click OK. (If Solver Add-in is not listed in the Add-Ins available box, click Browse to locate the add-in.)
5. If you get prompted that the Solver Add-in is not currently installed on your computer, click Yes to install it.

I have managed to make a VBA solution for this. I have tested with multiple different goals/targets & different range of numbers to sum and worked every time. Can't guarantee there isn't a problem it won't work around though.

Here it is:

Note - you should be able to now have multiple of the same number. This will only return the first solution it finds. It doesn't find EVERY solution.

Sub SumSolver()

Dim rng, Goal As Double, ws As Worksheet, i As Long, j As Long, Answer As Double, k As Long
Dim lRow As Long, Answerlist As String, LastAdded As Long, AnswerListPos As String
Dim c As Range, RngToSplit As String, AnswerArray, AnswerItem

Set ws = Sheets("Sheet1") 'Change Sheet1 to your sheet name
lRow = ws.Range("B" & ws.Rows.Count).End(xlUp).Row 'Change to needed column
ws.Range("C2:C" & lRow).ClearContents 'Clear output range if needed

For Each c In ws.Range("B2:B" & lRow) 'This loop populates the list range into a string
    If c.Value <> "" Or Not IsNumeric(c.Value) Then 'Checking for empty or non-numeric values
        If RngToSplit = "" Then
            RngToSplit = c.Value
        Else
            RngToSplit = RngToSplit & "," & c.Value
        End If
    End If
Next
rng = Split(RngToSplit, ",") 'Split the new list string into an array

If Not IsNumeric(ws.Range("A2").Value) Then 'Checks target value is actually a number
    MsgBox "The target value is not a valid number. Please correct this before trying again.", vbExclamation, "Sum Solver"
    Exit Sub
Else
    Goal = ws.Range("A2").Value 'Value of the goal/target
End If

For i = 0 To UBound(rng) ' 0 = start of array, Ubound = End of array
    If rng(i) = Goal Then
        ws.Range("C2") = rng(i)
        Answerlist = rng(i)
        GoTo SubExit
    ElseIf rng(i) < Goal Then
        Answer = rng(i)
        Answerlist = rng(i)
        AnswerListPos = i
        For j = i + 1 To UBound(rng)
            If Answer + rng(j) = Goal Then
                Answerlist = Answerlist & "," & rng(j)
                AnswerListPos = AnswerListPos & "," & j
                GoTo SubExit
            ElseIf Answer + rng(j) < Goal Then
                Answer = Answer + rng(j)
                LastAdded = j
                If Answerlist = "" Then
                    Answerlist = rng(j)
                    AnswerListPos = j
                Else
                    Answerlist = Answerlist & "," & rng(j)
                    AnswerListPos = AnswerListPos & "," & j
                End If
            End If
            If j = UBound(rng) Then
                If LastAdded = UBound(rng) Then
                    Answerlist = Left(Answerlist, InStrRev(Answerlist, ",") - 1)
                    AnswerListPos = Left(AnswerListPos, InStrRev(AnswerListPos, ",") - 1)
                    Answer = Answer - rng(j)
                    LastAdded = Val(Mid(AnswerListPos, InStrRev(AnswerListPos, ",") + 1))
                End If
                If LastAdded > 0 Then Answer = Answer - rng(LastAdded)
                If InStr(Answerlist, ",") = 0 Then Exit For
                j = Val(Mid(AnswerListPos, InStrRev(AnswerListPos, ",") + 1))
                Answerlist = Left(Answerlist, InStrRev(Answerlist, ",") - 1)
                AnswerListPos = Left(AnswerListPos, InStrRev(AnswerListPos, ",") - 1)
            End If
        Next j
    End If
    Answerlist = ""
Next i

SubExit:

If Answerlist <> "" Then
    i = 2
    AnswerArray = Split(Answerlist, ",") 'Split the result into an array
    For Each AnswerItem In AnswerArray
        ws.Range("C" & i) = AnswerItem 'Output the results into the sheet
        i = i + 1
    Next
Else
    MsgBox "No possible combination found for a target value of " & Goal & ".", vbExclamation, "Sum Solver"
End If

End Sub

EDIT: Just updated to account for if there are any blank rows in the list range as well as handle if a value is non-numeric. Actually made it half a second faster (13s) for a 12 item list of 1000 iterations.

You will see the rows I've made comments on are the ones you need to change. Pretty much just what columns it is working on and the starting cell It works looking at the last row but if you don't need it then just replace for example "B2:B" & lRow with B2:B5 etc.

I've also incorporated it into a function. Used as:

=SumSolver(Target value, Range of sum values)

It returns the results in the same cell separated by a comma. This can be changed to another method easily if needed though.

Function SumSolver(Goal As Double, ListRange As Range)

Dim i As Long, j As Long, Answer As Double, k As Long, rng As Variant
Dim Answerlist As String, LastAdded As Long, AnswerListPos As String

rng = Application.Transpose(ListRange)

For i = 1 To UBound(rng)
    If rng(i) = Goal Then
        Answerlist = rng(i)
        GoTo SubExit
    ElseIf rng(i) < Goal Then
        Answer = rng(i)
        Answerlist = rng(i)
        AnswerListPos = i
        For j = i + 1 To UBound(rng)
            If Answer + rng(j) = Goal Then
                Answerlist = Answerlist & "," & rng(j)
                AnswerListPos = AnswerListPos & "," & j
                GoTo SubExit
            ElseIf Answer + rng(j) < Goal Then
                Answer = Answer + rng(j)
                LastAdded = j
                If Answerlist = "" Then
                    Answerlist = rng(j)
                    AnswerListPos = j
                Else
                    Answerlist = Answerlist & "," & rng(j)
                    AnswerListPos = AnswerListPos & "," & j
                End If
            End If
            If j = UBound(rng) Then
                If LastAdded = UBound(rng) Then
                    Answerlist = Left(Answerlist, InStrRev(Answerlist, ",") - 1)
                    AnswerListPos = Left(AnswerListPos, InStrRev(AnswerListPos, ",") - 1)
                    Answer = Answer - rng(j)
                    LastAdded = Val(Mid(AnswerListPos, InStrRev(AnswerListPos, ",") + 1))
                End If
                If LastAdded > 0 Then Answer = Answer - rng(LastAdded)
                If InStr(Answerlist, ",") = 0 Then Exit For
                j = Val(Mid(AnswerListPos, InStrRev(AnswerListPos, ",") + 1))
                Answerlist = Left(Answerlist, InStrRev(Answerlist, ",") - 1)
                AnswerListPos = Left(AnswerListPos, InStrRev(AnswerListPos, ",") - 1)
            End If
        Next j
    End If
    Answerlist = ""
Next i

SubExit:

If Answerlist <> "" Then
    SumSolver = Answerlist
Else
    SumSolver = "N/A"
End If

End Function

Example:

I decided to do an update of my speed test. This time comparing the updated code against my original, and was surprised to see the difference. I did runs of 1000 iterations where the solver could not find a combination. I did this with screen updating on. For a list of 8 there is a max of 255 combinations, for a list of 12, there is a max of 4095 combinations (doubles with each item added). For a list of 12, that comes to 4,095,000 calculations. Updated code did that in an average of 13.6 seconds. Obviously this is on my machine, which will have different results to yours (but the ratios should still be about the same).