2024 AIME I Problem 4

Question

Jen enters a lottery by selecting

4

distinct elements of

S = \{1, 2, 3, \ldots, 9, 10\}.

Then four elements of

S

are drawn at random. Jen wins a prize if at least two of her numbers were drawn, and wins the grand prize if all four of her numbers were drawn. The probability that Jen wins the grand prize given that Jen wins a prize is

\frac{m}{n}

where

m

and

n

are relatively prime positive integers. Find

m + n.

Answer

Answer: 116

Concepts:conditional probability combinations

Difficulty rating: 2230

Small hint:

Winning the grand prize forces winning a prize, so the conditional probability is

\frac{P(\text{grand})}{P(\text{prize})}

Big hint:

Count draws that share exactly

2,

3,

or

4

numbers with Jen’s ticket — every draw of

4

numbers is equally likely

Solution:

All

\binom{10}{4} = 210

draws are equally likely. The number of draws sharing exactly

k

numbers with Jen’s ticket is

\binom{4}{k}\binom{6}{4-k},

so the number winning a prize is

\begin{aligned} &\binom{4}{2}\binom{6}{2} + \binom{4}{3}\binom{6}{1} \\ &\quad {}+ \binom{4}{4}\binom{6}{0} \\ &= 90 + 24 + 1 = 115, \end{aligned}

and exactly

1

of these wins the grand prize.

Since the grand prize implies a prize, the conditional probability is

\frac{\frac{1}{210}}{\frac{115}{210}} = \frac{1}{115},

so

m + n = 1 + 115 = 116.