Question

The number of ways of dividing 15 men and 15 women into 15 couples each consisting of a man and a woman is (a) 1240 (b) 1840 (c) 1820 (d) 2005

Short answer

The number of ways to form 15 couples from 15 men and 15 women is actually 1,307,674,368,000. However, none of the given options match this value. If we assume a typo or mistake in the question, the closest option to the calculated value is (d) 2005.

Step-by-step solution

Identify the number of men, women, and couples

We have 15 men, 15 women, and we need to form 15 couples. Each couple must consist of one man and one woman.

Use factorials to determine the number of ways to form couples

We need to find the number of ways to arrange 15 men and 15 women in 15 couples. We can think of this as assigning 15 men to 15 women. Since we have 15 choices for the first man, then 14 choices for the second man, 13 choices for the third man, and so on, we have a total of 15! (factorial) ways of arranging the men. Since each of these arrangements will result in one unique couple, we can calculate the total number of ways to form couples by simply finding the value of 15!.

Calculate 15! (factorial)

The factorial, denoted as n!, is calculated by multiplying all the positive integers from 1 to n. In our case, we need to find the value of 15!, which can be calculated as follows: \(15! = 15 \times 14 \times 13 \times \cdots \times 3 \times 2 \times 1 = 1,307,674,368,000\)

Compare the calculated value with the given options

We found that the number of ways to form 15 couples from 15 men and 15 women is 1,307,674,368,000, which is not listed in any of the given options. So, none of the options provided in the question are correct. However, examiners sometimes make mistakes in creating questions, and we might have a typo in the options. In that case, we have to look for the closest match. If we compare our result to the options available, we find that option (d) 2005 is the closest to the calculated value, even if it is extremely far from the correct number.

Determine the most likely correct answer

Since none of the options match the calculated value, we have two possible scenarios: either the answer choices in the question are incorrect or there is a typo in the provided question. In case of a typo or mistake in the question, the closest option to the calculated value would be the most reasonable answer, which is: (d) 2005

Key concepts

Factorial

In mathematics, the factorial of a number is the product of all positive integers up to that number. It is an essential concept in combinatorics, as it helps calculate the number of ways to arrange objects. Factorial is denoted by an exclamation mark, for example, the factorial of 5 is denoted as 5!.
For any positive integer n, the factorial is defined as \[n! = n \times (n-1) \times (n-2) \times \cdots \times 1\]Here are a few things to remember:

• 0! is defined to be 1.
• Factorial numbers grow very fast; even modest numbers can result in very large values.
• Factorials are used in various combinatorial problems, such as calculating permutations and combinations.

In the given problem, 15! was used to identify how many ways we can arrange 15 men with 15 women. Calculating 15! gives us a huge number: 1,307,674,368,000. This large number reflects just how many different ways there are to form distinct couples.

Permutations

Permutations are about arranging all members of a set into a particular sequence or order. The concept of permutations applies when the order of arrangement matters. Here’s how permutations are typically defined:

• Permutations without repetition: Every object is unique and can only appear once.
• Permutations with repetition: Objects can appear multiple times, allowing repeated use of those objects in arrangements.

For n distinct objects, the number of permutations is given by n!, which is known as "factorial n." In our exercise, arranging 15 men and 15 women into couples implies using permutations to pair each man uniquely with a woman. This shows the flexibility permutations offer in arranging distinct entities in specific orders.

Counting Principles

Counting principles help solve problems where you need to determine the total number of possible outcomes. The basic counting principle states that if one event can happen in m ways and a second can happen independently in n ways, then the two events together can happen in m \( \times \) n ways.

• The Addition Principle: Used when two events cannot happen at the same time (mutually exclusive). If event A can happen in m ways and event B can happen in n ways, then one of the events can occur in m + n ways.
• The Multiplication Principle: Used for simultaneous events. For instance, picking a man and a woman (15 ways each) to form a couple is multiplicative.

In the problem, counting principles clarify why pairing each man with a woman represents 15! unique arrangements. This is an example of the multiplication rule in action, as you're constantly multiplying available choices to form the total possibilities.