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Chapter 4: Problem 38

a. Five 鈥渕athletes鈥� celebrate after solving a particularly challenging problem during competition. If each mathlete high fives each other mathlete exactly once, what is the total number of high fives? b. If n mathletes shake hands with each other exactly once, what is the total number of handshakes? c. How many different ways can five mathletes be seated at a round table? (Assume that if everyone moves to the right, the seating arrangement is the same.) d. How many different ways can n mathletes be seated at a round table?

Short Answer

Expert verified
a. 10 high fives. b. \( \frac{n(n-1)}{2} \) handshakes. c. 24 ways. d. \( (n-1)! \) ways.

Step by step solution

01

Understanding part (a)

Each mathlete gives a high five to each other mathlete exactly once. This means that each pair of mathletes high fives each other.
02

Calculate total number of high fives in part (a)

To find the number of ways to choose 2 mathletes from 5, we use the combination formula \( nC2 = \frac{n!}{2!(n-2)!} \). For 5 mathletes, it is \( \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = 10 \)
03

Understanding part (b)

Each pair of mathletes shakes hands exactly once. For this, we again need to find the number of ways to choose 2 mathletes out of n.
04

Calculate total number of handshakes in part (b)

The number of ways for n mathletes to shake hands is given by \( nC2 = \frac{n!}{2!(n-2)!} \).
05

Understanding part (c)

Five mathletes seated at a round table is a permutation problem where rotations of the same arrangement are considered duplicates.
06

Calculate number of ways to seat 5 mathletes in part (c)

The number of unique arrangements is given by \( (5-1)! = 4! = 24 \).
07

Understanding part (d)

This is the general case for part (c), where we have n mathletes seated at a round table.
08

Calculate number of ways to seat n mathletes in part (d)

The number of unique arrangements of n mathletes in a round table is \( (n-1)! \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Combinations
A combination is a way of selecting items from a larger pool, where the order of selection does not matter. For example, if we need to choose 2 mathletes from a group of 5, we use combinations. The formula for combinations is \( nCk = \frac{n!}{k!(n-k)!} \). In our problem, we have 5 mathletes, and we need to find the number of ways any 2 of them can high-five. Using the combination formula, we get \( 5C2 = \frac{5!}{2!(5-2)!} = 10 \). This means there are 10 different pairs of mathletes who can high-five.
Permutations
Permutations are different from combinations in that the order of selection does matter. When arranging items or people, permutations are used. The general formula for permutations is \( P(n, k) = \frac{n!}{(n - k)!} \). In our problem, when seating mathletes at a round table, each different arrangement counts as a permutation. However, if rotations of the same circle are considered duplicates, the number of unique arrangements decreases. For example, for 5 mathletes seated at a round table, the number of unique permutations is \( (5-1)! = 4! = 24 \). This is because rotating the positions doesn't create a new arrangement.
Round Table Arrangement
Arranging people around a round table poses unique challenges as rotations of the same arrangement do not count as different. This implies that if you have n people, the number of unique circular permutations is (n-1)! because you can fix one person and arrange the remaining (n-1) people. For example, with 5 people, the number of unique seating arrangements is \( (5-1)! = 4! = 24 \). This ensures we do not count rotations as different. The general formula for the number of ways n people can be seated at a round table is \( (n-1)! \). This concept shows the special nature of circular permutations compared to linear arrangements.
Mathematical Competition
In the context of mathematical competitions, understanding combinatorics is crucial. Problems often involve scenarios like handshake exchanges or seating arrangements. For instance, if n mathletes shake hands exactly once, you use the combination formula \( nC2 \) to determine the number of pairs. Similarly, understanding permutations aids in solving problems where order matters, such as awards ceremonies or seating plans. Knowledge of these fundamental concepts ensures competitors can tackle a wide range of problems effectively. Competitions thus not only test mathematical skills but also enhance understanding of real-world applications of mathematics.

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