How many distinct distributions (x,y,z,w) of n
identical marbles in 4 boxes labeled A,B,C and D are there, such that
x,y,z,w are positive integers in strictly increasing order?
Verify the validity of your formula (or set of formulas) by manual listing of all such distributions for n=18.
In mathematical notation:
Sigma{x=1 to floor((n-1)/4)}
Sigma{y=x+1 to floor((n-1-x)/3)}
Sigma{z=y+1 to floor((n-1-x-y)/2)}
1
The algorithm produces
n distributions
10 1
11 1
12 2
13 3
14 5
15 6
16 9
17 11
18 15
19 18
20 23
21 27
22 34
23 39
24 47
25 54
26 64
27 72
28 84
29 94
30 108
31 120
32 136
33 150
34 169
35 185
36 206
37 225
38 249
39 270
40 297
41 321
42 351
43 378
44 411
45 441
46 478
47 511
48 551
49 588
50 632
51 672
52 720
53 764
54 816
55 864
56 920
57 972
58 1033
59 1089
60 1154
61 1215
62 1285
63 1350
64 1425
65 1495
66 1575
67 1650
68 1735
69 1815
70 1906
For n=18 the fifteen distributions are:
1 2 3 12
1 2 4 11
1 2 5 10
1 2 6 9
1 2 7 8
1 3 4 10
1 3 5 9
1 3 6 8
1 4 5 8
1 4 6 7
2 3 4 9
2 3 5 8
2 3 6 7
2 4 5 7
3 4 5 6
The program below uses a, b, c and d instead of x, y, z and w. The same idea. The for statement increments by 1 each time, so it will indeed stop with the floor of the terminal value given.
DefDbl A-Z
Dim crlf$
Private Sub Form_Load()
Form1.Visible = True
Text1.Text = ""
crlf = Chr$(13) + Chr$(10)
For n = 10 To 70
ct = 0
For a = 1 To (n - 1) / 4
For b = a + 1 To (n - a - 1) / 3
For c = b + 1 To (n - a - b - 1) / 2
d = n - a - b - c
ct = ct + 1
Next
Next
Next
Text1.Text = Text1.Text & n & Str(ct) & crlf
Next
Text1.Text = Text1.Text & crlf & " done"
End Sub
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Posted by Charlie
on 2017-08-16 11:41:37 |
solution
