Competition problem — Combinatorics

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Combinatorics Difficulty 4.7 AIME Find the answer United States

Problem:

Kelvin the frog jumps along the number line starting at $0$ . Every time he jumps, he jumps either one unit left or one unit right. For example, one sequence of jumps might be $0 \rightarrow -1 \rightarrow 0 \rightarrow 1 \rightarrow 2 \rightarrow 3 \rightarrow 2$ .
How many ways are there for Kelvin to make exactly $10$ jumps and land on a prime number? (The prime numbers are the sequence $2, 3, 5, 7, \ldots$ Negative numbers are not considered prime.)

Your answer

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

Solution

Solution:

First, note that every time Kelvin jumps, he must jump from an even number to an odd one or vice-versa. Thus after ten jumps, he must land on an even number. So, if that number is to be prime, it must be $2$ .

This means Kelvin must make $6$ jumps right and $4$ jumps left. That means the answer is $\binom{10}{6} = 210$ .

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