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Combinatorics Difficulty 4.0 AIME Find the answer United States

Problem:

The numbers $1, 2, \ldots, 10$ are written in a circle. There are four people, and each person randomly selects five consecutive integers (e.g. $1, 2, 3, 4, 5$ , or $8, 9, 10, 1, 2$ ). If the probability that there exists some number that was not selected by any of the four people is $p$ , compute $10000 p$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Solution:

The unselected numbers must be consecutive. Suppose that $\{1, 2, \ldots, k\}$ are the unselected numbers for some $k$ .

In this case, $1$ cannot be selected, so there are $5$ possible sets of consecutive numbers the people could have chosen. This leads to $5^{4}$ possibilities. Moreover, $10$ must be selected, so we must subtract $4^{4}$ possibilities where neither $1$ nor $10$ are selected.

Therefore, accounting for the rotation of the unselected numbers, we find
$p = \frac{10\left(5^{4} - 4^{4}\right)}{10^{4}} = \frac{3690}{10000}.$

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.