# i have a homework with 10 question but im stuck with 3 i searched about them everywhere read other c

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:
$\left(x+y{\right)}^{n}=\sum _{k=0}^{n}C\left(n,k\right)\ast {x}^{n-k}\ast {y}^{k}=\frac{{n}^{2}+n}{2}$
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.
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Kylan Simon
(2) Certainly we can write $1={2}^{0}\cdot 1$. Now suppose that $n>1$, and every positive integer less than n can be written in the desired form. (This is your induction hypothesis.) Then either $n$ is odd, or n is even. If $n$ is odd, we can write $n={2}^{0}\cdot n$ to express $n$ in the desired form. If $n$ is even, then $n=2m$ for some positive integer $m. By the induction hypothesis there are a non-negative integer $k$ and an odd integer $q$ such that $m={2}^{k}\cdot q$; how can you use this to express $n$ in the desired form? (This is not quite the proof that the author of the hint had in mind; it’s actually a bit easier.)

(6) As stated, this does not make sense. It’s true that
$\left(x+y{\right)}^{n}=\sum _{k=0}^{n}\left(\genfrac{}{}{0}{}{n}{k}\right){x}^{n-k}{y}^{k}\phantom{\rule{thickmathspace}{0ex}};$
this is the binomial theorem. It is not true that this equals $\frac{{n}^{2}+n}{2}$; this is obvious just from the fact that $\frac{{n}^{2}+n}{2}$ doesn’t depend on $x$ and $y$. What is true is that
$\sum _{k=0}^{n}k=\frac{{n}^{2}+n}{2}\phantom{\rule{thickmathspace}{0ex}};$
is this what you’re actually supposed to prove?
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agrejas0hxpx
(7) Let ${m}_{i}$ be the number of objects in the $i$-th box, and suppose that ${m}_{i}<{n}_{i}$ for $i=1,\dots ,t$. Then ${m}_{i}\le {n}_{i}-1$ for $i=1,\dots ,t$, so how big can $\sum _{i=1}^{t}{m}_{i}$ be?