5

Part of 2015 AIME Problems

Problems(2)

Drawing Pairs of Socks

Source: 2015 AIME I Problem 5

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 reaining 6 socks at random. The probability that Wednesday is the first day Sandy selects matching socks is

\tfrac{m}{n}

m

n

are relatively prime positive integers. Find

m+n

Probability of Two Selected Squares are Adjacent

Source: 2015 AIME 2 Problem 5

Two unit squares are selected at random without replacement from an

n\times n

grid of unit squares. Find the least positive integer

n

such that the probability that the two selected squares are horizontally or vertically adjacent is less than

\frac{1}{2015}