First hello all. i have a homework with 10 question but im stuck with 3 i searched about them everywhere read other colleges lectures but i couldnt solved them finally i desired to ask here
Question-2:
Prove that each positive integer can be written in form of $2^k*q$, where q is odd, and k is a non-negative integer.
Hint: Use induction, and the fact that the product of two odd numbers is odd.
Question-6:
$ (x+y)^n = \sum_{k=0}^n C(n,k)*x^{n-k}*y^k = {n^2+n\over 2} $ Prove the above statement by using induction on n.
Question-7:
Let $ n_1, n_2, ..., n_t $ be positive integers. Show that if $ n_1 + n_2 + ... + n_t - t + 1 $ objects are placed into $ t $ boxes, then for some i, $ i = 1, 2, 3, ... , t $ , the ith box contains at least $ n_i $ objects.
Your proof should not be more than 3 lines