Question

An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $7$ marbles of each color. When $7$ marbles are drawn randomly from the urn without replacement, the probability that exactly $4$ of them are red equals the probability that exactly $5$ of them are red. Find the sum of the five least values of $n$ for which this is possible.

答案

Solution:

Say there are $r$ red and $b$ blue marbles, $r, b \ge 7.$ The condition is $\binom{r}{4}\binom{b}{3} = \binom{r}{5}\binom{b}{2}.$ Since $\binom{r}{5} = \binom{r}{4}\frac{r-4}{5}$ and $\binom{b}{3} = \binom{b}{2}\frac{b-2}{3},$ cancelling gives $\frac{b - 2}{3} = \frac{r - 4}{5}, \qquad \text{that is,} \qquad b = \frac{3r - 2}{5}.$

So $r \equiv 4 \pmod 5,$ and $b \ge 7$ requires $3r - 2 \ge 35,$ so $r \ge 14.$ The five smallest choices are $r = 14, 19, 24, 29, 34$ with $b = 8, 11, 14, 17, 20,$ giving $n = 22, 30, 38, 46, 54.$

The sum is $22 + 30 + 38 + 46 + 54 = 190.$

2026 AIME II Problem 5

Solution:

Problem 5 in Other Years