We may decide to include or exclude the degenerate case of leaving one pile with all 5 black beads and all 5 white beads. It will be easiest to consider the possibilities of different numbers of piles separately. That brings up the question of whether a pile can have just one bead and still be considered a pile; I will assume you can have a pile with just one bead.

It's also the case that the piles don't have identities; they are not numbered boxes, so black beads in piles of 1, 2 and 2 is no different from being in piles of 2, 2 and 1. And if black beads occupy just three piles but white beads occupy four, there could be seven piles all together, or any number down to just four.

The hard part is combining the two lists; so much so, that I think the idea of separate lists is not a good idea, actually.

So, I think the best method is a recursive computer algorithm that reduces the number of available black and white beads as it creates piles, with number of white beads nested in a loop varying the number of black beads.

In the list below, each pile is represented by two digits, the first of which is the number of black beads and the second is the number of white beads.  The piles are listed in numeric order treating these as two-digit numbers.

 01 01 01 01 01 10 10 10 10 10

 01 01 01 01 01 10 10 10 20

 01 01 01 01 01 10 10 30

 01 01 01 01 01 10 20 20

 01 01 01 01 01 10 40

 01 01 01 01 01 20 30

 01 01 01 01 01 50

 01 01 01 01 10 10 10 10 11

 01 01 01 01 10 10 10 21

 01 01 01 01 10 10 11 20

 01 01 01 01 10 10 31

 01 01 01 01 10 11 30

 01 01 01 01 10 20 21

 01 01 01 01 10 41

 01 01 01 01 11 20 20

 01 01 01 01 11 40

 01 01 01 01 20 31

 01 01 01 01 21 30

 01 01 01 01 51

 01 01 01 02 10 10 10 10 10

 01 01 01 02 10 10 10 20

 01 01 01 02 10 10 30

 01 01 01 02 10 20 20

 01 01 01 02 10 40

 01 01 01 02 20 30

 01 01 01 02 50

 01 01 01 10 10 10 10 12

 01 01 01 10 10 10 11 11

 01 01 01 10 10 10 22

 01 01 01 10 10 11 21

 01 01 01 10 10 12 20

 01 01 01 10 10 32

 01 01 01 10 11 11 20

 01 01 01 10 11 31

 01 01 01 10 12 30

 01 01 01 10 20 22

 01 01 01 10 21 21

 01 01 01 10 42

 01 01 01 11 11 30

 01 01 01 11 20 21

 01 01 01 11 41

 01 01 01 12 20 20

 01 01 01 12 40

 01 01 01 20 32

 01 01 01 21 31

 01 01 01 22 30

 01 01 01 52

 01 01 02 10 10 10 10 11

 01 01 02 10 10 10 21

 01 01 02 10 10 11 20

 01 01 02 10 10 31

 01 01 02 10 11 30

 01 01 02 10 20 21

 01 01 02 10 41

 01 01 02 11 20 20

 01 01 02 11 40

 01 01 02 20 31

 01 01 02 21 30

 01 01 02 51

 01 01 03 10 10 10 10 10

 01 01 03 10 10 10 20

 01 01 03 10 10 30

 01 01 03 10 20 20

 01 01 03 10 40

 01 01 03 20 30

 01 01 03 50

 01 01 10 10 10 10 13

 01 01 10 10 10 11 12

 01 01 10 10 10 23

 01 01 10 10 11 11 11

 01 01 10 10 11 22

 01 01 10 10 12 21

 01 01 10 10 13 20

 01 01 10 10 33

 01 01 10 11 11 21

 01 01 10 11 12 20

 01 01 10 11 32

 01 01 10 12 31

 01 01 10 13 30

 01 01 10 20 23

 01 01 10 21 22

 01 01 10 43

 01 01 11 11 11 20

 01 01 11 11 31

 01 01 11 12 30

 01 01 11 20 22

 01 01 11 21 21

 01 01 11 42

 01 01 12 20 21

 01 01 12 41

 01 01 13 20 20

 01 01 13 40

 01 01 20 33

 01 01 21 32

 01 01 22 31

 01 01 23 30

 01 01 53

 01 02 02 10 10 10 10 10

 01 02 02 10 10 10 20

 01 02 02 10 10 30

 01 02 02 10 20 20

 01 02 02 10 40

 01 02 02 20 30

 01 02 02 50

 01 02 10 10 10 10 12

 01 02 10 10 10 11 11

 01 02 10 10 10 22

 01 02 10 10 11 21

 01 02 10 10 12 20

 01 02 10 10 32

 01 02 10 11 11 20

 01 02 10 11 31

 01 02 10 12 30

 01 02 10 20 22

 01 02 10 21 21

 01 02 10 42

 01 02 11 11 30

 01 02 11 20 21

 01 02 11 41

 01 02 12 20 20

 01 02 12 40

 01 02 20 32

 01 02 21 31

 01 02 22 30

 01 02 52

 01 03 10 10 10 10 11

 01 03 10 10 10 21

 01 03 10 10 11 20

 01 03 10 10 31

 01 03 10 11 30

 01 03 10 20 21

 01 03 10 41

 01 03 11 20 20

 01 03 11 40

 01 03 20 31

 01 03 21 30

 01 03 51

 01 04 10 10 10 10 10

 01 04 10 10 10 20

 01 04 10 10 30

 01 04 10 20 20

 01 04 10 40

 01 04 20 30

 01 04 50

 01 10 10 10 10 14

 01 10 10 10 11 13

 01 10 10 10 12 12

 01 10 10 10 24

 01 10 10 11 11 12

 01 10 10 11 23

 01 10 10 12 22

 01 10 10 13 21

 01 10 10 14 20

 01 10 10 34

 01 10 11 11 11 11

 01 10 11 11 22

 01 10 11 12 21

 01 10 11 13 20

 01 10 11 33

 01 10 12 12 20

 01 10 12 32

 01 10 13 31

 01 10 14 30

 01 10 20 24

 01 10 21 23

 01 10 22 22

 01 10 44

 01 11 11 11 21

 01 11 11 12 20

 01 11 11 32

 01 11 12 31

 01 11 13 30

 01 11 20 23

 01 11 21 22

 01 11 43

 01 12 12 30

 01 12 20 22

 01 12 21 21

 01 12 42

 01 13 20 21

 01 13 41

 01 14 20 20

 01 14 40

 01 20 34

 01 21 33

 01 22 32

 01 23 31

 01 24 30

 01 54

 02 02 10 10 10 10 11

 02 02 10 10 10 21

 02 02 10 10 11 20

 02 02 10 10 31

 02 02 10 11 30

 02 02 10 20 21

 02 02 10 41

 02 02 11 20 20

 02 02 11 40

 02 02 20 31

 02 02 21 30

 02 02 51

 02 03 10 10 10 10 10

 02 03 10 10 10 20

 02 03 10 10 30

 02 03 10 20 20

 02 03 10 40

 02 03 20 30

 02 03 50

 02 10 10 10 10 13

 02 10 10 10 11 12

 02 10 10 10 23

 02 10 10 11 11 11

 02 10 10 11 22

 02 10 10 12 21

 02 10 10 13 20

 02 10 10 33

 02 10 11 11 21

 02 10 11 12 20

 02 10 11 32

 02 10 12 31

 02 10 13 30

 02 10 20 23

 02 10 21 22

 02 10 43

 02 11 11 11 20

 02 11 11 31

 02 11 12 30

 02 11 20 22

 02 11 21 21

 02 11 42

 02 12 20 21

 02 12 41

 02 13 20 20

 02 13 40

 02 20 33

 02 21 32

 02 22 31

 02 23 30

 02 53

 03 10 10 10 10 12

 03 10 10 10 11 11

 03 10 10 10 22

 03 10 10 11 21

 03 10 10 12 20

 03 10 10 32

 03 10 11 11 20

 03 10 11 31

 03 10 12 30

 03 10 20 22

 03 10 21 21

 03 10 42

 03 11 11 30

 03 11 20 21

 03 11 41

 03 12 20 20

 03 12 40

 03 20 32

 03 21 31

 03 22 30

 03 52

 04 10 10 10 10 11

 04 10 10 10 21

 04 10 10 11 20

 04 10 10 31

 04 10 11 30

 04 10 20 21

 04 10 41

 04 11 20 20

 04 11 40

 04 20 31

 04 21 30

 04 51

 05 10 10 10 10 10

 05 10 10 10 20

 05 10 10 30

 05 10 20 20

 05 10 40

 05 20 30

 05 50

 10 10 10 10 15

 10 10 10 11 14

 10 10 10 12 13

 10 10 10 25

 10 10 11 11 13

 10 10 11 12 12

 10 10 11 24

 10 10 12 23

 10 10 13 22

 10 10 14 21

 10 10 15 20

 10 10 35

 10 11 11 11 12

 10 11 11 23

 10 11 12 22

 10 11 13 21

 10 11 14 20

 10 11 34

 10 12 12 21

 10 12 13 20

 10 12 33

 10 13 32

 10 14 31

 10 15 30

 10 20 25

 10 21 24

 10 22 23

 10 45

 11 11 11 11 11

 11 11 11 22

 11 11 12 21

 11 11 13 20

 11 11 33

 11 12 12 20

 11 12 32

 11 13 31

 11 14 30

 11 20 24

 11 21 23

 11 22 22

 11 44

 12 12 31

 12 13 30

 12 20 23

 12 21 22

 12 43

 13 20 22

 13 21 21

 13 42

 14 20 21

 14 41

 15 20 20

 15 40

 20 35

 21 34

 22 33

 23 32

 24 31

 25 30

 55

 

339 269   70 done

Meaning there are 339 ways, but 269 of these involve at least one "pile" that is only a single bead, leaving 70 that consist solely of piles, strictly speaking.

DefDbl A-Z

Dim crlf$, pile(10), blackremain, whiteremain, tot, totsingle

Private Sub Form_Load()

 Text1.Text = ""

 crlf$ = Chr(13) + Chr(10)

 Form1.Visible = True

 

 blackremain = 5: whiteremain = 5

 addOn 1

 

 Text1.Text = Text1.Text & crlf & tot & Str(totsingle) & "   " & tot - totsingle & " done"

End Sub

Sub addOn(wh)

  If wh = 1 Then blackstart = 0 Else blackstart = pile(wh - 1) \ 10

  For black = blackstart To blackremain

    blackremain = blackremain - black

    

    For white = 0 To whiteremain

     DoEvents

      whiteremain = whiteremain - white

      

      pile(wh) = 10 * black + white

      If pile(wh) > 0 And pile(wh) >= pile(wh - 1) Then

        If blackremain = 0 And whiteremain = 0 Then

          oneOnly = 0

          For i = 1 To wh

            Text1.Text = Text1.Text & Format(pile(i), " 00")

            If pile(i) = 1 Or pile(i) = 10 Then oneOnly = 1

          Next

          Text1.Text = Text1.Text & crlf

          tot = tot + 1

          If oneOnly Then totsingle = totsingle + 1

        Else

          addOn wh + 1

        End If

      End If

    

      whiteremain = whiteremain + white

    Next white

    

    blackremain = blackremain + black

  Next black

End Sub