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**Aquafina** Let $\displaystyle \binom{n}{r}$ stand for the number of subsets of size r taken from a set of size n. (This is the number of ways of choosing r objects from n if the order does not matter.) Every subset of the set 1, 2, . . . , n either contains the element 1 or it doesn’t. By considering these two

possibilities, show that:

$\displaystyle \binom{n-1}{r-1} + \binom{n-1}{r} = \binom{n}{r}$

By using a similar method or otherwise, prove that:

$\displaystyle \binom{n-2}{r-2} + 2\binom{n-2}{r-1} + \binom{n-2}{r} = \binom{n}{r}$

I'm not sure what to do. I do not understand the queston, with what they said in the 1st paragraph and how to apply it here. Can i basically use the idea of binomial coefficients and factorials, and simplify the fraction or what?

Thanks for the help