Problem 159 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

How many different words can be formed by jumbling the letters in the word MISSISSIPPI in which no two S are adjacent? (A) \(8 \cdot{ }^{6} C_{4} \cdot{ }^{7} C_{4}\) (B) \(6 \cdot 7 \cdot{ }^{8} C_{4}\) (C) \(6 \cdot 8 \cdot{ }^{7} C_{4}\) (D) \(7 \cdot{ }^{6} C_{4} \cdot{ }^{8} C_{4}\)

Step-by-Step Solution

Answer

The correct answer is (C) \(6 \cdot 8 \cdot { }^{7}C_{4}\).

Step 1: Count Total Letters

The word 'MISSISSIPPI' consists of 11 letters, where there are 4 S's, 4 I's, 2 P's, and 1 M.

Step 2: Arrange Non-S Letters

First, arrange the non-S letters (M, I, I, I, I, P, P), which total 7 letters. The number of permutations for arranging them is given by \( \frac{7!}{4!2!} \) to account for repeating I's and P's.

Step 3: Separate S's

To ensure no two S's are adjacent, place S's in gaps created by arranging the non-S letters. The arrangement of 7 non-S letters results in 8 gaps (one before, one between each letter, and one after).

Step 4: Choose Gaps for S's

Choose 4 gaps out of the 8 available to place the S's, which is calculated as \( \binom{8}{4} \).

Step 5: Combine Arrangements

Combine the arrangements calculated in Step 2 and Step 4, resulting in the total number of permutations: \[ \frac{7!}{4!2!} \times \binom{8}{4} \].

Step 6: Evaluate Expression

Evaluate the mathematical expression \( \frac{7!}{4!2!} \times \binom{8}{4} = 105 \times 70 = 7350 \).

Key Concepts

Permutations and Combinations

Permutations and combinations are fundamental concepts in combinatorics, and they are used to count different ways of arranging or choosing items from a set. A permutation refers to an arrangement of objects in a specific order. For example, if you want to know how many different ways you can arrange the letters in the word "CAT", you would calculate the permutations of the three letters, which is 3! = 6.

• Permutations: Concerned with the arrangement.
• Formula: For n different objects, the number of permutations of n is n!.

Combinations, on the other hand, focus on the selection of objects without regard to order. For example, if you have 5 books and want to choose 2 to take on a trip, you are looking for the number of combinations. This is given by \( \binom{n}{r} \), where n is the total number of items, and r is the number to choose.

• Combinations: Concerned with the selection.
• Formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Factorial

The factorial of a number is a crucial mathematical concept used in various fields, especially combinatorics. It is used to represent the product of an integer and all the non-zero integers below it. For example, the factorial of 5 (denoted 5!) is 5 × 4 × 3 × 2 × 1 = 120.

• Definition: \( n! = n \times (n-1) \times \cdots \times 2 \times 1 \)
• Special Case: By convention, 0! is defined as 1.

Factorials are used in permutations and combinations to calculate possible arrangements and selections of a set. Understanding how to manipulate factorials is fundamental when solving problems involving large numbers.
Factorials grow very quickly. For instance, 10! is already 3,628,800, which helps explain why permutations and combinations can yield such large numbers of arrangements.

Binomial Coefficient

The binomial coefficient, often denoted as \( \binom{n}{r} \), is a key concept in combinatorics for calculating combinations. It signifies the number of ways to choose r elements from a set of n elements, without regard to order. This concept is extensively used in probability, statistics, and algebra.

• Notation: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)
• Application: Often used to calculate probabilities and in expressions of the binomial theorem.

In the problem of arranging letters in the word MISSISSIPPI with specific conditions, the binomial coefficient helps to decide how many ways we can pick particular gaps to place certain letters, ensuring rules such as non-adjacency are met. For instance, choosing 4 gaps from 8 entirely relies on correctly using the binomial coefficient formula.