Reduce summation to expression

Asked Nov 11 '22 at 16:30

Active Nov 11 '22 at 16:30

Viewed 27 times

I am working on a probability problem and ran across the following sum...

$$ \sum_{i=1}^{k} {n-i\choose r} $$

where $k$ is some arbitrary positive integer such that $k \le n-r-1$.

is there a simple, compact expression/formula for it?

I noticed that the expansion consists of pairs of terms that almost permit the application of Pascal's rule, but the terms are all positive so this is not an option.

I am also aware of the hockey stick identity mentioned here, which would be useful if the sum was from $i=1,2,...,k,...n-r-1$, but the sum is only over terms up to some arbitrary $k$.

Thanks for any feedback!

asked Nov 11 '22 at 16:30

RyRy the Fly Guy

5,950
1
11
27

1

Write it as $$ \left[\sum_{i=0}^{n-1}\binom{i}r\right]- \left[\sum_{i=0}^{n-k-1}\binom{i}r\right],$$ then use the hockey stick identity. – Mike Earnest Nov 11 '22 at 16:35

Reduce summation to expression

0 Answers0