Question

(16 choose 0) + (16 choose 1) + ..... + (16 choose 16)
Please Help!

Answer 1

Use the binomial theorem:

$(1+1)^{16}=\displaystyle\sum_{k=0}^{16}\binom{16}k1^{16-k}1^k$

So

$\dbinom{16}0+\dbinom{16}1+\cdots+\dbinom{16}{16}=\boxed{2^{16}}$

More generally,

$\displaystyle\sum_{k=0}^n\binom nk=2^n$