Question

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 groups that can be selected. What is the remainder when $N$ is divided by $100?$

$47$

$48$

$83$

$95$

$96$

Solution:

Choosing $t$ tenors and $b$ basses is weighted by $\binom{6}{t}\binom{8}{b}.$ To keep only $t - b \equiv 0 \pmod 4,$ apply a roots of unity filter with $\omega = i:$ $N + 1 = \frac14\sum_{j=0}^{3}(1 + i^{j})^6\,(1 + i^{-j})^8.$

The $j = 0$ term is $2^6\cdot 2^8 = 16384.$ The $j = 2$ term has factor $(1 + i^2)^6 = 0.$ The $j = 1$ and $j = 3$ terms are $-128i$ and $128i,$ which cancel. So the sum is $16384,$ and $\dfrac{16384}{4} = 4096.$

This count includes the empty group, so $N = 4096 - 1 = 4095,$ and $N \equiv 95 \pmod{100}.$

Thus, the correct answer is D.

2021 AMC 12A Spring Problem 15

Solution:

Problem 15 in Other Years