Difference between revisions of "2021 AMC 12A Problems/Problem 15"

Revision as of 11:53, 12 February 2021

Problem

A choir direction must select a group of singers from among his $6$ tenors and $8$ basses. The only requirements are that the difference between the number of tenors and basses must be a multiple of $4$ , and the group must have at least one singer. Let $N$ be the number of different groups that could be selected. What is the remainder when $N$ is divided by $100$ ?

$\textbf{(A) } 47\qquad\textbf{(B) } 48\qquad\textbf{(C) } 83\qquad\textbf{(D) } 95\qquad\textbf{(E) } 96\qquad$

Video Solution by Punxsutawney Phil

https://youtube.com/watch?v=FD9BE7hpRvg&t=533s

Solution 2

We know the choose function and we know the pair multiplication $MN$ so we do the multiplications and additions. $\binom{6}{0}(\binom{8}{4}+\binom{8}{8})+\binom{6}{1}(\binom{8}{1}+\binom{8}{5})+\binom{6}{2}(\binom{8}{2}+\binom{8}{6})+\binom{6}{3}(\binom{8}{3}+\binom{8}{7})+\binom{6}{4}(\binom{8}{0}+\binom{8}{4}+\binom{8}{8})+\binom{6}{5}(\binom{8}{1}+\binom{8}{5})+\binom{6}{6}(\binom{8}{2}+\binom{8}{6}) = \boxed{(D) 4095}$ ~Lopkiloinm

Video Solution by Hawk Math

https://www.youtube.com/watch?v=AjQARBvdZ20

Video Solution by OmegaLearn (using Vandermonde's Identity)

https://www.youtube.com/watch?v=mki7xtZLk1I

~pi_is_3.14

Solution 5: Generating Functions

The problem can be done using a roots of unity filter. Let $f(x,y)=(1+x)^8(1+y)^6$ . By expanding the binomials and distributing, $f(x,y)$ is the generating function for different groups of basses and tenors. That is, $f(x,y)=\sum_{m=0}^8\sum_{n=0}^6 a_{mn}x^my^n$ where $a_{mn}$ is the number of groups of $m$ basses and $n$ tenors. What we want to do is sum up all values of $a_{mn}$ for which $4\mid |m-n|$ except for $a_{00}=1$ . To do this, define a new function $g(x)=f(x,x^{-1})=\sum_{m=0}^8\sum_{n=0}^6 a_{mn}x^{m-n}=(1+x)^8(1+x^{-1})^6.$ Now we just need to sum all coefficients of $g(x)$ for which $4\mid |m-n|$ . Consider a monomial $h(x)=x^k$ . If $4\mid k$ , $h(i)+h(-1)+h(-i)+h(1)=1+1+1+1=4$ otherwise, $h(i)+h(-1)+h(-i)+h(1)=0.$ $g(x)$ is a sum of these monomials so this gives us a method to determine the sum we're looking for: $\frac{g(i)+g(-1)+g(-i)+g(1)}{4}=2^{12}=4096$ (since $g(-1)=0$ and it can be checked that $g(i)=-g(-i)$ ). Hence, the answer is $4096-1$ with the $-1$ for $a_{00}$ which gives $\boxed{95}$ . ~lawliet163

@@ Line 42: / Line 42: @@
 \frac{g(i)+g(-1)+g(-i)+g(1)}{4}=2^{12}=4096
 </cmath>
-(since <math>g(-1)=0</math> and <math>g(i)=-g(-1)</math>). Hence, the answer is <math>4096-1</math> with the <math>-1</math> for <math>a_{00}</math> which gives <math>\boxed{95}</math>.
+(since <math>g(-1)=0</math> and it can be checked that <math>g(i)=-g(-i)</math>). Hence, the answer is <math>4096-1</math> with the <math>-1</math> for <math>a_{00}</math> which gives <math>\boxed{95}</math>.
 ~lawliet163

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
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