Question

A man has 4 sons. There are 6 proposals of marriage for his sons. In how many ways can they select a proposal for their marriages, such that none of them marry with more than one girl? (a) 180 (b) 270 (c) 360 (d) none of these

Short answer

Answer: There are 360 ways for the 4 sons to select a proposal for their marriages without marrying more than one girl.

Step-by-step solution

Identify the problem type

This is a combinatorial problem involving permutations. We need to find the number of ways to distribute the 6 proposals among the 4 sons, without any son getting more than one proposal.

Calculate the total number of permutations

We will apply the formula for permutations to find the total number of ways to distribute the 6 proposals among the 4 sons. The formula for permutations is given by: P(n, r) = n! / (n-r)! Where n is the total number of items (proposals), r is the number of items to be selected (sons), and ! denotes the factorial function. In this problem, n = 6 (proposals) and r = 4 (sons).

Calculate the factorials

Next, we need to calculate the factorials for n and (n-r): n! = 6! = 6 × 5 × 4 × 3 × 2 × 1 = 720 (n-r)! = (6-4)! = 2! = 2 × 1 = 2

Calculate the permutations

Now, we can use the factorials to find the permutations: P(6, 4) = 720 / 2 = 360

State the final answer

There are 360 ways for the 4 sons to select a proposal for their marriages, such that none of them marry with more than one girl. The correct answer is (c) 360.

Key concepts

Factorial Function

The factorial function, symbolized by an exclamation mark (!), is essential in combinatorics, particularly when we calculate permutations and combinations. The factorial of a non-negative integer, say \( n \), is the product of all positive integers less than or equal to \( n \). For example, \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

Special cases of the factorial function include \( 0! \), which is defined as 1, and this convention is particularly useful in combinatorial problems to ensure consistent results even when there are no items to permute or combine.

It is the foundation upon which the permutation formula is built, as it counts the number of ways to arrange a given number of objects when the order is important. This concept becomes vividly clear as we explore permutations next.

Permutation Formula

In combinatorics, the permutation formula calculates the number of possible arrangements of a set of items where the order is significant. The general permutation formula is \( P(n, r) = \frac{n!}{(n-r)!} \) where \( n \) is the total number of items and \( r \) is the number to be chosen from \( n \).

For example, if you have 5 books and wish to arrange 3 of them on a shelf, you would use the permutation formula to find the number of different arrangements possible. In the context of our exercise, the permutation formula is used to determine how many ways the 4 sons can receive marriage proposals from a pool of 6.

Understanding this formula is crucial as it helps in organizing items into a sequence, considering each unique order as a separate case. In real-world scenarios, permutations help solve problems related to scheduling, assigning tasks, and organizing events where the sequence of occurrences matters.

Combinatorial Problems

Combinatorial problems are puzzles or questions that involve finding the number of ways certain arrangements or selections can be made. These problems are common in mathematics, statistics, and computer science. They involve understanding the nature of the problem to select the correct technique among permutations, combinations, or other combinatorial principles.

A common example is the problem of distributing objects among recipients—like in the original exercise where marriage proposals must be assigned to sons in such a way that none receives more than one proposal. The factorial function and permutation formula are tools that make solving these combinatorial problems possible.

Mastering combinatorial problem-solving is not only academically rewarding but also has practical applications in various fields like cryptography, research design, game theory, and even in everyday tasks like organizing a closet or planning a seating arrangement for a dinner party.