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 ${2}^{k}\ast q$, 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:

$(x+y{)}^{n}=\sum _{k=0}^{n}C(n,k)\ast {x}^{n-k}\ast {y}^{k}=\frac{{n}^{2}+n}{2}$

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.

Question-2:

Prove that each positive integer can be written in form of ${2}^{k}\ast q$, 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:

$(x+y{)}^{n}=\sum _{k=0}^{n}C(n,k)\ast {x}^{n-k}\ast {y}^{k}=\frac{{n}^{2}+n}{2}$

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.