### ADVANCED EXERCISES - Expanding Binomials

**(1)** Use the Binomial Theorem to show that

n + n + n + ... + n + n = 2^{n}. 0 1 2 n-1 n

**(2)**Use the Binomial Theorem to show that

n - n + n - ... + (-1)^{k }n + ... + (-1)^{n }n = 0 . 0 1 2 k n

**(3)**Use the Binomial Coefficient Formula and 0! = 1 to prove the following identities:

(i) n = 1 . 0 (ii) n = n . 1 (iii) n = n for 0 r n . r n-r