help with binomial theorem identities and probability questions?

Question

prove the following identities: (n choose 0) - (n choose 1) + (n choose 2) -(n choose 3) + .......(-1)^n (n choose n) = 0

(n choose 0) + (n choose 2) + (n choose 4) + (n choose 6) +.......(n choose n)= 2^n-1 if n is even.

probability questions: A bridge hand consists of 13 randomly chosen cards from 52. What's the probability that a bridge hand consists of 4 clubs, 4 diamonds, 3 spades, and 2 hearts?

i got (13 choose 4)(13 choose 4)(13 choose 3)(13 choose 2) all divided by (52 choose 4). which i got to equal 1.795%. Not sure if that is right.

If a bridge hand consists of 4 clubs, 4 diamonds, 3 spades, and 2 hearts. Whats the prob. that they happen to fall in this order?

I was thinking (13 C 4) / (52 C 4) times (13 C 4) / (52 C 4) times (13 C 3) / (52 C 4) times (13 C 2) / (52 C 4). I dont think i did this right at all, but i could be wrong.

Thanks a lot for any help or suggestions.

Answer 1

1) write decomposition of (1-1)^n=0
2) write decomposition of (1+1)^n = 2^n. Add the result with the decomposition in 1)

Answer 2

I'm not even going to ask what the big word is. ^_^

Answer 3

I don't understand your terminology.