I have been trying to prove a formula of mine and i come across something very interesting, well to me it is. If the formula is correct, it states that: $$ \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_1 \choose v_1}(-1)^{k-c-m}\right)\left(\sum_{m=0}^{c} {c \choose m}{m(a-s_1) \choose v_2}\right) = \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_2 \choose v_1}(-1)^{k-c-m}\right)\left(\sum_{m=0}^{c} {c \choose m}{m(a-s_2) \choose v_2}\right) $$
With $k\leq v_1 \leq l $, $c \leq v_1 \leq n-l$,$0 \leq c \leq n$, $c \leq k \leq n$, $s_1 \ne s_2$, and $a$ can be a number of your choice that is greater than $s_1$ and $s_2$
So my question is of course, does the formula really work? how can i prove if the formula works or does not work
We talked earlier on RS about this, and I mentioned that a good way to approach this sort of challenge is to try and show that the statement is wrong. If you can easily do this, it will save you a lot of time and frustration (since otherwise you'd be trying to prove something that isn't true).
Anyways, following up on our discussion, try using k = 1, c = 0, l = 3, n = 6, v1 = 2, v2 = 0, a = 10, s1 = 1, s2 = 2. This satisfies the bounds listed, and it will yield the statement 1 = 0 which is false.