High Fives


b. If n mathletes shake hands with each other exactly once, what is the total number of handshakes?

Verified step by step guidance

1. Understand the problem: The goal is to calculate the total number of handshakes if n mathletes each shake hands with every other mathlete exactly once. This is a combinatorics problem where we need to count the number of unique pairs that can be formed from n individuals.
2. Recall the formula for combinations: The number of ways to choose 2 individuals from a group of n is given by the combination formula: $\frac{n!}{r!(n-r)!}$, where n is the total number of individuals, r is the size of the group being chosen (in this case, 2), and ! denotes factorial.
3. Simplify the formula for this specific case: Since r = 2, the formula becomes $\frac{n(n-1)}{2}$. This represents the total number of unique pairs (handshakes) that can be formed.
4. Substitute the value of n into the formula: Replace n with the given number of mathletes in the problem. For example, if there are 5 mathletes, substitute n = 5 into the formula.
5. Perform the calculation: Compute the value of $\frac{n(n-1)}{2}$ to find the total number of handshakes. This will give you the final answer.

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with combinations and arrangements of objects. In the context of the handshake problem, it helps determine how many unique pairs can be formed from a group of 'n' individuals. The formula used is based on combinations, specifically 'n choose 2', which calculates the number of ways to select 2 individuals from 'n' without regard to the order.

Combination Formula

The combination formula, denoted as C(n, k) = n! / (k!(n-k)!), is used to find the number of ways to choose 'k' elements from a set of 'n' elements. For the handshake problem, we set k to 2, as each handshake involves a pair of mathletes. This formula simplifies the calculation of handshakes by eliminating the need to list all possible pairs.

Factorial

A factorial, denoted as n!, is the product of all positive integers up to 'n'. It is a fundamental concept in combinatorics, as it provides the basis for calculating combinations and permutations. In the handshake problem, factorials are used in the combination formula to compute the total number of unique handshakes among 'n' mathletes, ensuring accurate counting of pairs.