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

Revision as of 10:35, 24 August 2021

Problem

A choir director 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 $100$ ?

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

Solution 1 (Bijection)

Let $t$ be the number of tenors selected and $b$ be the number of basses selected. The requirements are $t\equiv b\pmod{4}$ and $(t,b)\neq(0,0).$

It follows that $b'=8-b$ is the number of basses not selected. Since the ordered pairs $(t,b)$ and the ordered pairs $(t,b')$ have one-to-one correspondence, we consider the ordered pairs $(t,b')$ instead. The requirements become $t\equiv 8-b'\pmod{4}$ and $(t,8-b')\neq(0,0),$ which simplify to $t+b'\equiv8\pmod{4}$ and $(t,b')\neq(0,8),$ respectively.

Solution 2 (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 $4\mid k$ , then $h(i)+h(-1)+h(-i)+h(1)=1+1+1+1=4.$ 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 $4096-1=4095\equiv\boxed{\textbf{(D) } 95}\pmod{100}$ .

~lawliet163

Solution 3 (Casework and Vandermonde's Identity)

By casework, we construct the following table. In the last column, we rewrite some of the combinations using the identity $\binom{n}{r}=\binom{n}{n-r}:$ $\begin{array}{c|c|c|c} & & & \\ [-2ex] \textbf{\# of Tenors} & \textbf{\# of Basses} & \textbf{\# of Groups} & \textbf{Rewrite \# of Groups} \\ [0.5ex] \hline\hline & & & \\ [-2ex] 0 & 8 & \tbinom{6}{0}\tbinom{8}{8} & \\ [1ex] 1 & 1 & \tbinom{6}{1}\tbinom{8}{1} & \tbinom{6}{1}\tbinom{8}{7}\\ [1ex] 2 & 2 & \tbinom{6}{2}\tbinom{8}{2} & \tbinom{6}{2}\tbinom{8}{6}\\ [1ex] 3 & 3 & \tbinom{6}{3}\tbinom{8}{3} & \tbinom{6}{3}\tbinom{8}{5}\\ [1ex] 4 & 4 & \tbinom{6}{4}\tbinom{8}{4} & \\ [1ex] 5 & 5 & \tbinom{6}{5}\tbinom{8}{5} & \tbinom{6}{5}\tbinom{8}{3}\\ [1ex] 6 & 6 & \tbinom{6}{6}\tbinom{8}{6} & \tbinom{6}{6}\tbinom{8}{2}\\ [1ex] \hline & & & \\ [-2ex] 0 & 4 & \tbinom{6}{0}\tbinom{8}{4} & \\ [1ex] 1 & 5 & \tbinom{6}{1}\tbinom{8}{5} & \tbinom{6}{1}\tbinom{8}{3}\\ [1ex] 2 & 6 & \tbinom{6}{2}\tbinom{8}{6} & \tbinom{6}{2}\tbinom{8}{2}\\ [1ex] 3 & 7 & \tbinom{6}{3}\tbinom{8}{7} & \tbinom{6}{3}\tbinom{8}{1}\\ [1ex] 4 & 8 & \tbinom{6}{4}\tbinom{8}{8} & \tbinom{6}{4}\tbinom{8}{0}\\ [1ex] \hline & & & \\ [-2ex] 4 & 0 & \tbinom{6}{4}\tbinom{8}{0} & \tbinom{6}{2}\tbinom{8}{0}\\ [1ex] 5 & 1 & \tbinom{6}{5}\tbinom{8}{1} & \tbinom{6}{1}\tbinom{8}{1}\\ [1ex] 6 & 2 & \tbinom{6}{6}\tbinom{8}{2} & \tbinom{6}{0}\tbinom{8}{2}\\ [1ex] \end{array}$ We apply Vandermonde's Identity to find the total number of such groups: $\begin{align*} N&=\underbrace{\left[\sum_{k=0}^{6}\binom{6}{k}\binom{8}{8-k}\right]}_{\tbinom{14}{8}}+\underbrace{\left[\sum_{k=0}^{4}\binom{6}{k}\binom{8}{4-k}\right]}_{\tbinom{14}{4}}+\underbrace{\left[\sum_{k=0}^{2}\binom{6}{k}\binom{8}{2-k}\right]}_{\tbinom{14}{2}} \\ &=\binom{14}{6}+\binom{14}{4}+\binom{14}{2} \\ &=3003+1001+91 \\ &=4095 \\ &\equiv\boxed{\textbf{(D) } 95}\pmod{100}. \end{align*}$ ~MRENTHUSIASM

Solution 4 (Enumeration)

By casework, we construct the following table: $\begin{array}{c|c|c|c} & & & \\ [-2ex] \textbf{\# of Tenors} & \textbf{\# of Basses} & \textbf{\# of Groups} & \textbf{Evaluate \# of Groups} \\ [0.5ex] \hline\hline & & & \\ [-2ex] 0 & 4 & \tbinom{6}{0}\tbinom{8}{4} & 70 \\ [1ex] 0 & 8 & \tbinom{6}{0}\tbinom{8}{8} & 1 \\ [1ex] \hline & & & \\ [-2ex] 1 & 1 & \tbinom{6}{1}\tbinom{8}{1} & 48 \\ [1ex] 1 & 5 & \tbinom{6}{1}\tbinom{8}{5} & 336 \\ [1ex] \hline & & & \\ [-2ex] 2 & 2 & \tbinom{6}{2}\tbinom{8}{2} & 420 \\ [1ex] 2 & 6 & \tbinom{6}{2}\tbinom{8}{6} & 420 \\ [1ex] \hline & & & \\ [-2ex] 3 & 3 & \tbinom{6}{3}\tbinom{8}{3} & 1120 \\ [1ex] 3 & 7 & \tbinom{6}{3}\tbinom{8}{7} & 160 \\ [1ex] \hline & & & \\ [-2ex] 4 & 0 & \tbinom{6}{4}\tbinom{8}{0} & 15 \\ [1ex] 4 & 4 & \tbinom{6}{4}\tbinom{8}{4} & 1050 \\ [1ex] 4 & 8 & \tbinom{6}{4}\tbinom{8}{8} & 15 \\ [1ex] \hline & & & \\ [-2ex] 5 & 1 & \tbinom{6}{5}\tbinom{8}{1} & 48 \\ [1ex] 5 & 5 & \tbinom{6}{5}\tbinom{8}{5} & 336 \\ [1ex] \hline & & & \\ [-2ex] 6 & 2 & \tbinom{6}{6}\tbinom{8}{2} & 28 \\ [1ex] 6 & 6 & \tbinom{6}{6}\tbinom{8}{6} & 28 \\ [1ex] \end{array}$ We find the total number of such groups: $70+1+48+336+420+420+1120+160+15+1050+15+48+336+28+28=4095\equiv\boxed{\textbf{(D) } 95}\pmod{100}.$ ~sugar_rush

~MRENTHUSIASM

Video Solution by Punxsutawney Phil

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

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

@@ Line 5: / Line 5: @@
 ==Solution 1 (Bijection)==
-<b>WILL FINISH BY TOMORROW. NO EDIT PLEASE.</b>
+Let <math>t</math> be the number of tenors selected and <math>b</math> be the number of basses selected. The requirements are <math>t\equiv b\pmod{4}</math> and <math>(t,b)\neq(0,0).</math>
+It follows that <math>b'=8-b</math> is the number of basses not selected. Since the ordered pairs <math>(t,b)</math> and the ordered pairs <math>(t,b')</math> have one-to-one correspondence, we consider the ordered pairs <math>(t,b')</math> instead. The requirements become <math>t\equiv 8-b'\pmod{4}</math> and <math>(t,8-b')\neq(0,0),</math> which simplify to <math>t+b'\equiv8\pmod{4}</math> and <math>(t,b')\neq(0,8),</math> respectively.
 ==Solution 2 (Generating Functions)==
-The problem can be done using a roots of unity filter. Let <math>f(x,y)=(1+x)^8(1+y)^6</math>. By expanding the binomials and distributing, <math>f(x,y)</math> is the generating function for different groups of basses and tenors. That is,
+The problem can be done using a roots of unity filter. Let <math>f(x,y)=(1+x)^8(1+y)^6</math>. By expanding the binomials and distributing, <math>f(x,y)</math> is the generating function for different groups of basses and tenors. That is, <cmath>f(x,y)=\sum_{m=0}^8\sum_{n=0}^6 a_{mn}x^my^n,</cmath>
-<cmath>f(x,y)=\sum_{m=0}^8\sum_{n=0}^6 a_{mn}x^my^n,</cmath>
+where <math>a_{mn}</math> is the number of groups of <math>m</math> basses and <math>n</math> tenors. What we want to do is sum up all values of <math>a_{mn}</math> for which <math>4\mid m-n</math> except for <math>a_{00}=1</math>. To do this, define a new function <cmath>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.</cmath>
-where <math>a_{mn}</math> is the number of groups of <math>m</math> basses and <math>n</math> tenors. What we want to do is sum up all values of <math>a_{mn}</math> for which <math>4\mid m-n</math> except for <math>a_{00}=1</math>. To do this, define a new function
+Now we just need to sum all coefficients of <math>g(x)</math> for which <math>4\mid m-n</math>. Consider a monomial <math>h(x)=x^k</math>. If <math>4\mid k</math>, then <cmath>h(i)+h(-1)+h(-i)+h(1)=1+1+1+1=4.</cmath>
-<cmath>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.</cmath>
-Now we just need to sum all coefficients of <math>g(x)</math> for which <math>4\mid m-n</math>. Consider a monomial <math>h(x)=x^k</math>. If <math>4\mid k</math>,
-<cmath>h(i)+h(-1)+h(-i)+h(1)=1+1+1+1=4.</cmath>
 Otherwise,
 <cmath>h(i)+h(-1)+h(-i)+h(1)=0.</cmath>

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 14	Followed by Problem 16
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
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