whether is $\sum_{k=0}^{j-1} \binom{i}{k}=\sum_{k=0}^{j-2} \binom{i-1}{k}+\sum_{k=0}^{j-1} \binom{i-1}{k}$ true or false?

Question

I have tested some trivial samples when $j = 1,2,3$. But I can't prove if it is true or false generally. Any help would be great, thanks!

score 0 · Accepted Answer · answered May 21 '16 at 14:00

0

Use the fact that $${i\choose k}={i-1\choose k}+{i-1\choose k-1}$$ , which can be proved by expanding the terms.

answered May 21 '16 at 14:00

Emre

1

Or combinatorially: all groups of $k$ from $i$ either contain the max of the set (last term) or not (first term). – Henno Brandsma May 21 '16 at 14:02

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