I am a high school student and I am searching for a not very complicated way to calculate $${100\choose 0}^2-{100\choose 1}^2+{100\choose 2}^2-{100\choose 3}^2+...+{100\choose 100}^2.$$
6864
I know that the sum of squares of binomial coefficients is just ${2n}\choose{n}$ but what is the closed expression for the sum ${n\choose 0}^2 - {n\choose 1}^2 + {n\choose 2}^2 + \cdots + (-1)^n {n\choose n}^2$?
68078
How can we find the prime factorization of $100!$. My only idea is to write it as,  $$(1)(2)(3)(\color{red}{2})(\color{red}2)...$$  And count the factors.  But that would take way to long. Any ideas?
102061
Question :  What is the remainder when   $$ 10^{10} + 10^{10^2} +10^{10^3} + \ldots + 10^{10^{100}} $$  is divided by $7$?
76258
What will be the remainder obtained when the polynomial $x^{100}$ is divided by the polynomial $(x-2)(x-1)$.  I used remainder theorem but it had no impact in the solution.
65848
$(1+px+qx^2)^8 \equiv  1+8x+52x^2+kx^3...$  Given that expression, calculate the values of p, q, and k.  I'm guessing this is a trinomial that needs to be turned into a binomial, but how would you do it? I have no idea.
124091
No. of ways to choose $n$ things from $n$ alike "a" things $n$ alike "b" things and $n$ different things.  Answer is $(n+2)2^{n-1}$. But how to prove it..?
59458
Not sure what to do / how to start this... I have equcation of 504 is: $2 \cdot2 \cdot 2 \cdot 3 \cdot 3 \cdot 7$
64693
I actually got this part and I got $16$ choose $2$, which would be $120$. The part I didn't get which wouldn't fit into the title was in how many of these solutions is $x_1 \geq 1$, $x_2 \geq 2$,and $x_3 \geq 3$? I'm not very good at combinatorics so I don't really know. I think the way to approach it would be saying how many solutions where $x &lt; 1 +$ solutions where $x &lt; 2 +$ solutions where $x &lt; 3$ and subtract that from $120$, but I don't know how to do that.
35858
Prove that if 100 numbers are chosen from the first 200 natural   numbers and include a number less than 16, then one of them is   divisible by another.   How to prove this? many thanks....
114427
The coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)...(1+x^{100})$.  I tried the following concept, how to sum 9 using 1-9 only without repetition.  1:9, 2:1+8,3:7+2, 4: 6+3, 5: 5+4,  6:1+2+6 ,7:1+3+5, 8:2+3+4  The answer is 8.  How will it be solved using Binomial theorem.
73819
I'm trying to find the last two digits of ${2012}^{2012}$. I know you can use (mod 100) to find them, but I'm not quite sure how to apply this. Can someone please explain it?
37573
$\left(x-\displaystyle\frac{x^3}{6}+\displaystyle\frac{x^5}{120} +o(x^5)\right)\left(1+\displaystyle\frac{x^2}{2}-\displaystyle\frac{x^4}{24}+ \displaystyle\frac{x^4}{8}+o(x^4)\right) = x+\displaystyle\frac{x^3}{3}+\displaystyle\frac{2x^5}{15}+o(x^5)$
31453
I'm well aware of the combinatorial variant of the proof, i.e. noting that each formula is a different representation for the number of subsets of a set of $n$ elements. I'm curious if there's a series of algebraic manipulations that can lead from $\sum\limits_{i=0}^n \binom{n}{i}$ to $2^n$.
96204
I'm looking for some guidance in solving this. I have read through my text multiple times trying to find something similar to this and I'm coming up with nothing. I'm not really sure where to start on solving this problem.  The question simply states     Prove that ${{n}\choose{k}} = {{n - 2}\choose{k}} + 2{{n - 2}\choose{k - 1}} + {{n - 2}\choose{k - 2}}$.   I'm very new to binomial coefficients, and extremely lost.
89595
How many $0's$ are in the end of:  $$1^1 \cdot 2^2 \cdot 3^3 \cdot 4^4.... 99^{99}$$  The answer is supposed to be $1100$ but I have absolutely NO clue how to get there.  Any advice?
92829
I am trying to solve the following question which is Ex3 from Arthur Engel, Problem Solving strategies. Here is the question: $\sum_{k=1}^n k^3 {n \choose k}$ and asks to find the sum. I am sincerely having a hard time trying to figure out how to solve it and get $n^2 (n+3)(2)^{n-3}$. I would really appreciate if someone could show the step-by-step process. Thank you!
15986
Please help me calculate the following sum     $$\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+\dots+\frac{1}{99\cdot 100}$$