Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2015 based on Probability.

## Probability Problem – AIME I, 2015

In a drawer Sandy has 5 pairs of socks, each pair a different color. on monday sandy selects two individual socks at random from the 10 socks in the drawer. On tuesday Sandy selects 2 of the remaining 8 socks at random and on wednesday two of the remaining 6 socks at random. The probability that wednesday is the first day Sandy selects matching socks is \(\frac{m}{n}\), where m and n are relatively prime positive integers, find m+n.

- is 107
- is 341
- is 840
- cannot be determined from the given information

**Key Concepts**

Algebra

Theory of Equations

Probability

## Check the Answer

Answer: is 341.

AIME, 2015, Question 5

Geometry Revisited by Coxeter

## Try with Hints

Wednesday case – with restriction , select the pair on wednesday in \(5 \choose 1 \) ways

Tuesday case – four pair of socks out of which a pair on tuesday where a pair is not allowed where 4 pairs are left,the number of ways in which this can be done is \(8 \choose 2\) – 4

Monday case – a total of 6 socks and a pair not picked \(6 \choose 2\) -2

by multiplication and principle of combinatorics \(\frac{(5)({5\choose 2} -4)({6 \choose 2}-2)}{{10 \choose 2}{8 \choose 2}{6 \choose 2}}\)=\(\frac{26}{315}\). That is 341.

## Other useful links

- https://www.cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s