Questions:

a)Suppose that r and n are integers with $\displaystyle 1 \le r \le n$.

Show that $\displaystyle ^n\mathrm{C}_r + ^n\mathrm{C}_{r-1} = ^{n+1}\mathrm{C}_r$

b)If n is a positive integer,

Show that $\displaystyle 2^n= ^n\mathrm{C}_0 + ^n\mathrm{C}_1 + ^n\mathrm{C}_2 +......+ ^n\mathrm{C}_{n-1} + ^n\mathrm{C}_n $

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My attempt so far:

a)$\displaystyle ^n\mathrm{C}_r + ^n\mathrm{C}_{r-1} = \frac{n!}{(n-r)!r!} + \frac{n!}{(n-(r-1))!(r-1)!} $

$\displaystyle ^n\mathrm{C}_r + ^n\mathrm{C}_{r-1} = \frac{n! (n-r+1)!(r-1)! + n!(n-r)!r!}{(n-r)!r!(n-r+1)!(r-1)!}$

Ugh! Any ideas where I could go from here?

b)$\displaystyle 2^n = \sum (^n\mathrm{C}_n) = \sum \frac{n!}{(n-n)!n!} = \sum \frac{n!}{n!} = \sum 1 $

which is clearly wrong.

Please help! Thank you!