Track / Stage 5 / 82 of 400 #682 of 1964

Problem 682

AIME late

Combinatorics Difficulty 5.2 Find the answer

8. Six boys and $a$ girls stand in a row at random, with each boy being adjacent to at least one other boy, and the probability that at least four boys stand together is $p$ . If $p \leqslant \frac{1}{100}$ , then the minimum value of $a$ is . $\qquad$

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Official solution

8.594.

Notice that,
$\begin{aligned} p & =\frac{2 \mathrm{C}_{a+1}^{2}+(a+1)}{\mathrm{C}_{a+1}^{3}+3 \mathrm{C}_{a+1}^{2}+(a+1)} \\ & =\frac{6 a+6}{a^{2}+8 a+6} \leqslant \frac{1}{100} . \end{aligned}$

Therefore, $f(a)=a^{2}-592 a-594 \geqslant 0$ .
Also, $f(0)0$ , thus, the minimum value of $a$ is 594.

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.