Problem 89 - A Complete Resource Book in Mathematics for JEE Main

Question

The total number of ways of selecting five letters from the letters of the word INDEPENDENT, is (A) 4200 (B) 3320 (C) 3840 (D) None of these

Step-by-Step Solution

Answer

46 ways, so the answer is (D) None of these.

Step 1: Identify the Letters and their Counts

In the word 'INDEPENDENT', the distinct letters are: I, N (3 times), D (2 times), E (3 times), P, T. The counts of each are: I=1, N=3, D=2, E=3, P=1, T=1.

Step 2: Apply Combinatorics Principles

Since some letters repeat, we need to consider combinations that account for this repetition. The problem can be solved by considering cases based on how many times we select each letter.

Step 3: Case Analysis

We consider different cases based on the repetition: 1) Choose two N's and two E's, and one more letter from remaining. 2) Choose three N's, two of another letter. Analyze each combination's possibilities separately.

Step 4: Calculate for the Case: Two N's and Two E's, Plus One Other

Choose 2 N's from 3 \((^3C_2)\), then choose 2 E's from 3 \((^3C_2)\), then choose 1 additional letter from the remaining: I, D, or P, or T \((^4C_1)\). The calculation: \[^3C_2 \times ^3C_2 \times ^4C_1 = 3 \times 3 \times 4 = 36\]

Step 5: Calculate for the Case: Three N's, Plus Two Other

Choose 3 N's from 3 \((^3C_3)\), then choose 2 letters from the set of distinct letters \{I, D, E, P, T\}, resulting in \(^5C_2\). The calculation: \[^3C_3 \times ^5C_2 = 1 \times 10 = 10\]

Step 6: Compile the Total

Add all the combinations from possible cases to get the total number: \(36 + 10 = 46\).

Step 7: Final Decision

Verify if 46 matches any options. Since it doesn’t match, the answer is 'None of these'.

Key Concepts

Combinations with Repetition

Combinations with repetition allow us to count how many ways we can select items when the order doesn’t matter and items can be repeated.
In the context of the original exercise, the concept applies when we are selecting five letters from the word "INDEPENDENT." Repeated letters make this a perfect candidate for combinations with repetition.

• Since some letters appear more than once in "INDEPENDENT," careful case-based analysis is needed. Cases were based on selecting letters multiple times, like choosing multiple N’s or E’s.
• Each distinct combination takes repetition into account, ensuring that each selection's uniqueness is maintained while honoring how many times each letter can appear.

Thinking this way ensures we cover all possible selections, even when specific letters repeat.

Permutations and Combinations

Permutations and combinations are two fundamental sub-topics in combinatorics, each serving unique purposes.
Permutations deal with ordered arrangements, while combinations are concerned with selections where order doesn't matter.

• In solving the exercise from the original problem, combinations were pivotal because the order of selecting letters does not matter.
• Instead, the focus was on how to select a group of letters respecting repetition constraints.
• Combinatorial formulas like \({}^nC_r\) are applied to identify how many ways letters can be chosen, considering repeats.

The understanding of these concepts enables systematic approaches in selecting or arranging different entities, as required in a given problem.

Mathematical Problem Solving

Mathematical problem solving involves approaching issues using structured methods and logical reasoning.
This strategy is key to handling complex puzzles like the one from this exercise, which required breaking problems into manageable steps.

• First, identify constraints such as which letters repeat and how often they can be used. This sets the problem's parameters.
• Next, create a plan by breaking the problem into smaller cases and solving each case independently to ensure all scenarios are considered.
• Finally, compile solutions from each case to form a total solution, verifying against given options to ensure accuracy or find new insights.

This strategic approach is valuable not just in combinatorics, but in broader educational and real-world problem-solving contexts.