In how many ways can you write 20 as a sum of three counting numbers?
One Way: 1+18+1
Not a Way: 1+1+1+17
Anybody now how to solve this?
I assume in your case, there is only a recursive method:
Let P(n,k) is the number of partitions of a positive integer n into exactly k parts:
For instance, \(7 = 5+1+1=4+2+1=3+2+2\), so \(p(7,3) = 4\)
Formula:
\(\begin{array}{|rcll|} \hline P(0,0) &=& 1 \\ P(n,0) &=& 0 \qquad n\ge 1 \\ P(n,1 )&=& 1 \\ P(n,n) &=& 1 \\ P(n,k) &=& P(n-k,k)+P(n-1,k-1) \qquad \text{ or } \qquad P(n+k,k) = \sum \limits_{j=1}^{k}P(n,j) \\ \hline \end{array} \)
In how many ways can you write 20 as a sum of three counting numbers?
p(20,3) = 33
P(n,k):
p(n,k):
n = 1 -------------------------
p(1,1) = 1
n = 2 -------------------------
p(2,1) = 1
p(2,2) = 1
n = 3 -------------------------
p(3,1) = 1
p(3,2) = 1
p(3,3) = 1
n = 4 -------------------------
p(4,1) = 1
p(4,2) = 2
p(4,3) = 1
p(4,4) = 1
n = 5 -------------------------
p(5,1) = 1
p(5,2) = 2
p(5,3) = 2
p(5,4) = 1
p(5,5) = 1
n = 6 -------------------------
p(6,1) = 1
p(6,2) = 3
p(6,3) = 3
p(6,4) = 2
p(6,5) = 1
p(6,6) = 1
n = 7 -------------------------
p(7,1) = 1
p(7,2) = 3
p(7,3) = 4
p(7,4) = 3
p(7,5) = 2
p(7,6) = 1
p(7,7) = 1
n = 8 -------------------------
p(8,1) = 1
p(8,2) = 4
p(8,3) = 5
p(8,4) = 5
p(8,5) = 3
p(8,6) = 2
p(8,7) = 1
p(8,8) = 1
n = 9 -------------------------
p(9,1) = 1
p(9,2) = 4
p(9,3) = 7
p(9,4) = 6
p(9,5) = 5
p(9,6) = 3
p(9,7) = 2
p(9,8) = 1
p(9,9) = 1
n = 10 -------------------------
p(10,1) = 1
p(10,2) = 5
p(10,3) = 8
p(10,4) = 9
p(10,5) = 7
p(10,6) = 5
p(10,7) = 3
p(10,8) = 2
p(10,9) = 1
p(10,10) = 1
n = 11 -------------------------
p(11,1) = 1
p(11,2) = 5
p(11,3) = 10
p(11,4) = 11
p(11,5) = 10
p(11,6) = 7
p(11,7) = 5
p(11,8) = 3
p(11,9) = 2
p(11,10) = 1
p(11,11) = 1
n = 12 -------------------------
p(12,1) = 1
p(12,2) = 6
p(12,3) = 12
p(12,4) = 15
p(12,5) = 13
p(12,6) = 11
p(12,7) = 7
p(12,8) = 5
p(12,9) = 3
p(12,10) = 2
p(12,11) = 1
p(12,12) = 1
n = 13 -------------------------
p(13,1) = 1
p(13,2) = 6
p(13,3) = 14
p(13,4) = 18
p(13,5) = 18
p(13,6) = 14
p(13,7) = 11
p(13,8) = 7
p(13,9) = 5
p(13,10) = 3
p(13,11) = 2
p(13,12) = 1
p(13,13) = 1
n = 14 -------------------------
p(14,1) = 1
p(14,2) = 7
p(14,3) = 16
p(14,4) = 23
p(14,5) = 23
p(14,6) = 20
p(14,7) = 15
p(14,8) = 11
p(14,9) = 7
p(14,10) = 5
p(14,11) = 3
p(14,12) = 2
p(14,13) = 1
p(14,14) = 1
n = 15 -------------------------
p(15,1) = 1
p(15,2) = 7
p(15,3) = 19
p(15,4) = 27
p(15,5) = 30
p(15,6) = 26
p(15,7) = 21
p(15,8) = 15
p(15,9) = 11
p(15,10) = 7
p(15,11) = 5
p(15,12) = 3
p(15,13) = 2
p(15,14) = 1
p(15,15) = 1
n = 16 -------------------------
p(16,1) = 1
p(16,2) = 8
p(16,3) = 21
p(16,4) = 34
p(16,5) = 37
p(16,6) = 35
p(16,7) = 28
p(16,8) = 22
p(16,9) = 15
p(16,10) = 11
p(16,11) = 7
p(16,12) = 5
p(16,13) = 3
p(16,14) = 2
p(16,15) = 1
p(16,16) = 1
n = 17 -------------------------
p(17,1) = 1
p(17,2) = 8
p(17,3) = 24
p(17,4) = 39
p(17,5) = 47
p(17,6) = 44
p(17,7) = 38
p(17,8) = 29
p(17,9) = 22
p(17,10) = 15
p(17,11) = 11
p(17,12) = 7
p(17,13) = 5
p(17,14) = 3
p(17,15) = 2
p(17,16) = 1
p(17,17) = 1
n = 18 -------------------------
p(18,1) = 1
p(18,2) = 9
p(18,3) = 27
p(18,4) = 47
p(18,5) = 57
p(18,6) = 58
p(18,7) = 49
p(18,8) = 40
p(18,9) = 30
p(18,10) = 22
p(18,11) = 15
p(18,12) = 11
p(18,13) = 7
p(18,14) = 5
p(18,15) = 3
p(18,16) = 2
p(18,17) = 1
p(18,18) = 1
n = 19 -------------------------
p(19,1) = 1
p(19,2) = 9
p(19,3) = 30
p(19,4) = 54
p(19,5) = 70
p(19,6) = 71
p(19,7) = 65
p(19,8) = 52
p(19,9) = 41
p(19,10) = 30
p(19,11) = 22
p(19,12) = 15
p(19,13) = 11
p(19,14) = 7
p(19,15) = 5
p(19,16) = 3
p(19,17) = 2
p(19,18) = 1
p(19,19) = 1
n = 20 -------------------------
p(20,1) = 1
p(20,2) = 10
p(20,3) = 33
p(20,4) = 64
p(20,5) = 84
p(20,6) = 90
p(20,7) = 82
p(20,8) = 70
p(20,9) = 54
p(20,10) = 42
p(20,11) = 30
p(20,12) = 22
p(20,13) = 15
p(20,14) = 11
p(20,15) = 7
p(20,16) = 5
p(20,17) = 3
p(20,18) = 2
p(20,19) = 1
p(20,20) = 1
...
In how many ways can you write 20 as a sum of three counting numbers?
p(20,3) = 33
\(\begin{array}{|r|ll|} \hline & 20 = \\ \hline 1 & 1+1+18 \\ 2 & 1+2+17 \\ 3 & 1+3+16 \\ 4 & 1+4+15 \\ 5 & 1+5+14 \\ 6 & 1+6+13 \\ 7 & 1+7+12 \\ 8 & 1+8+11 \\ 9 & 1+9+10 \\ \hline 10 & 2+2+16 \\ 11 & 2+3+15 \\ 12 & 2+4+14 \\ 13 & 2+5+13 \\ 14 & 2+6+12 \\ 15 & 2+7+11 \\ 16 & 2+8+10 \\ 17 & 2+9+9 \\ \hline 18 & 3+3+14 \\ 19 & 3+4+13 \\ 20 & 3+5+12 \\ 21 & 3+6+11 \\ 22 & 3+7+10 \\ 23 & 3+8+9 \\ \hline 24 & 4+4+12 \\ 25 & 4+5+11 \\ 26 & 4+6+10 \\ 27 & 4+7+9 \\ 28 & 4+8+8 \\ \hline 29 & 5+5+10 \\ 30 & 5+6+9 \\ 31 & 5+7+8 \\ \hline 32 & 6+6+8 \\ 33 & 6+7+7 \\ \hline \end{array}\)