2021 AMC 12A Problems/Problem 15

Revision as of 23:34, 17 May 2021 by MRENTHUSIASM (talk | contribs) (Solution 4 (Combinatorial Argument))

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 (Without Words)

\[\begin{array}{c|c|c} \text{Tenors} & \text{Basses} & \text{Groups} \\  \hline 0 & 4, 8 & \binom{6}{0}\left[\binom{8}{4}+\binom{8}{8}\right]=70+1=\fbox{71} \\ 1 & 1, 5 & \binom{6}{1}\left[\binom{8}{1}+\binom{8}{5}\right]=6(8+56)=6(64)=3\fbox{84} \\ 2 & 2, 6 & \binom{6}{2}\left[\binom{8}{2}+\binom{8}{6}\right]=15(28+28)=15(56)=8\fbox{40} \\ 3 & 3, 7 & \binom{6}{3}\left[\binom{8}{3}+\binom{8}{7}\right]=20(56+8)=20(64)=12\fbox{80} \\ 4 & 0, 4, 8 & \binom{6}{4}\left[\binom{8}{0}+\binom{8}{4}+\binom{8}{8}\right]=15(1+70+1)=15(72)=10\fbox{80} \\ 5 & 1, 5 & \binom{6}{5}\left[\binom{8}{1}+\binom{8}{5}\right]=6(8+56)=6(64)=3\fbox{84} \\ 6 & 2, 6 & \binom{6}{6}\left[\binom{8}{2}+\binom{8}{6}\right]=1(28+28)=\fbox{56}  \end{array}\]

$71+384+840+1280+1080+384+56=40\boxed{\textbf{(D)} ~95}$

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 $4\mid k$, \[h(i)+h(-1)+h(-i)+h(1)=1+1+1+1=4\] 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

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 $\binom{n}{r}=\binom{n}{n-r}:$ \[\begin{array}{c|c|c|c} & & & \\ [-2ex] \textbf{\# of Tenors} & \textbf{\# of Basses} & \textbf{\# of Ways} & \textbf{Rewrite \# of Ways} \\ [0.5ex] \hline\hline  & & & \\ [-2ex] 0 & 8 & \tbinom{6}{0}\tbinom{8}{8} & \\ [1ex] 1 & 1 & \tbinom{6}{1}\tbinom{8}{1} & \tbinom{6}{1}\tbinom{8}{7}\\ [1ex] 2 & 2 & \tbinom{6}{2}\tbinom{8}{2} & \tbinom{6}{2}\tbinom{8}{6}\\ [1ex] 3 & 3 & \tbinom{6}{3}\tbinom{8}{3} & \tbinom{6}{3}\tbinom{8}{5}\\ [1ex] 4 & 4 & \tbinom{6}{4}\tbinom{8}{4} & \\ [1ex] 5 & 5 & \tbinom{6}{5}\tbinom{8}{5} & \tbinom{6}{5}\tbinom{8}{3}\\ [1ex] 6 & 6 & \tbinom{6}{6}\tbinom{8}{6} & \tbinom{6}{6}\tbinom{8}{2}\\ [1ex] \hline & & & \\ [-2ex] 0 & 4 & \tbinom{6}{0}\tbinom{8}{4} & \\ [1ex] 1 & 5 & \tbinom{6}{1}\tbinom{8}{5} & \tbinom{6}{1}\tbinom{8}{3}\\ [1ex] 2 & 6 & \tbinom{6}{2}\tbinom{8}{6} & \tbinom{6}{2}\tbinom{8}{2}\\ [1ex] 3 & 7 & \tbinom{6}{3}\tbinom{8}{7} & \tbinom{6}{3}\tbinom{8}{1}\\ [1ex] 4 & 8 & \tbinom{6}{4}\tbinom{8}{8} & \tbinom{6}{4}\tbinom{8}{0}\\ [1ex] \hline & & & \\ [-2ex] 4 & 0 & \tbinom{6}{4}\tbinom{8}{0} & \tbinom{6}{2}\tbinom{8}{0}\\ [1ex] 5 & 1 & \tbinom{6}{5}\tbinom{8}{1} & \tbinom{6}{1}\tbinom{8}{1}\\ [1ex] 6 & 2 & \tbinom{6}{6}\tbinom{8}{2} & \tbinom{6}{0}\tbinom{8}{2}\\ [1ex] \end{array}\] We apply Vandermonde's Identity to find the requested sum: \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 (Combinatorial Argument)

We claim that if the empty group is allowed, then there are $\mathbf{2^{12}}$ valid ways to choose the singers.

First, we set one tenor and one bass aside. We argue that each group from the $12$ remaining singers (of any size, including $0$) corresponds to exactly one valid group from the $14$ original singers.

The $12$ remaining singers can form \[2^{12}=\left[\sum_{t=0}^{5}\binom5t\right]\cdot\left[\sum_{b=0}^{7}\binom7b\right]\] groups. The left side counts directly, while the right side uses casework (selecting $t$ tenors and $b$ basses for each group). Now, we map each group from the remaining $12$ to one valid group from the original $14.$

By casework:

  1. $|b-t|\equiv1\mathrm{ \ or \ }3\pmod{4}$
  2. Clearly, the mapping is satisfied. For each group from the remaining $12,$ we can obtain one valid group from the original $14$ by adding one tenor or one bass accordingly.

  3. $|b-t|\equiv0\pmod{4}$
  4. Since $\binom7b=\binom{7}{7-b},$ we can select $7-b$ basses instead of $b$ basses, without changing the number of groups.

    The absolute difference becomes $\left|(7-b)-t\right|=\left|7-(b+t)\right|.$ Since $b-t$ is even, we conclude that $b+t$ must also be even. It follows that $\left|(7-b)-t\right|=\left|7-(b+t)\right|\equiv 1\mathrm{ \ or \ }3\pmod{4}.$ This mapping is satisfied by Case 1.

  5. $|b-t|\equiv2\pmod{4}$
  6. By the same reasoning as Case 2, we select $7-b$ basses instead of $b$ basses.

    The absolute difference also becomes $\left|(7-b)-t\right|=\left|7-(b+t)\right|\equiv 1\mathrm{ \ or \ }3\pmod{4}.$ This mapping is satisfied by Case 1.

Now, the proof of the bolded claim is complete.

Therefore, excluding the empty group gives $N=2^{12}-1=4095\equiv\boxed{\textbf{(D) } 95}\pmod{100}.$

~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

See also

2021 AMC 12A (ProblemsAnswer KeyResources)
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
All AMC 12 Problems and Solutions

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