Combinatorics : number of non-decreasing series of r distinct numbers where the size of series ranges from 1 to N

Asked Apr 05 '15 at 12:56

Active Apr 05 '15 at 12:56

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I understand that number of non decreasing sequences of size M with N distinct numbers is (N+M−1)C(M). However, I'm interested in finding out the number of such series of r distinct integers where the size ranges from 1 to N. So its summation of (N+M−1)C(M) where N=1 to N. Is there a way I can directly get the answer without summing. A formula or a pattern that may apply ?

-Aman

asked Apr 05 '15 at 12:56

Aman Jha

Possible duplicate of How to simplify $\sum_{i=1}^{k}\binom{n + i - 1}{i}$? – Prasun Biswas Apr 05 '15 at 12:59
Your question isn’t entirely clear. Suppose that $r=2$, you have the two numbers $1$ and $2$, and $N=3$; are $1$, $2$, $11$, $12$, $22$, $111$, $112$, $122$, and $222$ the sequences that you want to count? – Brian M. Scott Apr 05 '15 at 20:57
If so, then @Prasun is correct, and you should look at that link. – Brian M. Scott Apr 05 '15 at 21:05

Combinatorics : number of non-decreasing series of r distinct numbers where the size of series ranges from 1 to N

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