Problem 55 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

From 6 ditferent novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on the shelf so that the dictionary is always in the middle. Then the number of such arrangements is (A) less than 500 (B) at least 500 but less than 750 (C) at least 750 but less than 1000 (D) at least 1000

Step-by-Step Solution

Answer

(D) at least 1000.

Step 1: Select the Novels

First, we need to select 4 novels from the 6 available. The number of ways to choose 4 novels from 6 is given by the combination formula: \( \binom{6}{4} \). Calculating this gives us \( \binom{6}{4} = 15 \).

Step 2: Select the Dictionary

We need to select 1 dictionary from the 3 available ones. The number of ways to choose 1 dictionary from 3 is \( \binom{3}{1} \). Calculating this gives us \( \binom{3}{1} = 3 \).

Step 3: Arrange the Books on the Shelf

Once the books are selected, they must be arranged so that the dictionary is in the middle. This means there is one fixed position for the dictionary, requiring us to arrange the 4 novels on either side of it. The number of arrangements of the 4 novels is given by the factorial \( 4! \). Calculating this gives us \( 4! = 24 \).

Step 4: Calculate Total Arrangements

Multiply the number of ways to choose the novels, choose the dictionary, and arrange the novels. The total number of arrangements is \( 15 \times 3 \times 24 = 1080 \).

Step 5: Interpret the Total Number

Based on the calculated arrangements, we determine which option category this number falls into. Since 1080 is at least 1000, the correct choice corresponds to option (D).

Key Concepts

Factorial Calculation

Factorial calculation is a fundamental concept of permutations in mathematics. When arranging items, understanding factorials helps determine how many possible arrangements there are for a set number of items. The factorial of a positive integer, denoted by the symbol \( n! \), is the product of all positive integers less than or equal to \( n \).

For example, \( 4! \) means multiplying 4 by every integer below it: 4, 3, 2, and 1. This results in:

• \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)

This calculation is crucial in determining the number of ways to arrange novels around a fixed position dictionary in our original exercise.

Factorials grow very quickly, which makes them useful for calculating possible arrangements in problems involving larger sets. Understanding factorials makes the concept of permutations and combinations much more accessible for solving complex problems.

Combination Formula

The combination formula is used to determine how many ways you can select items from a larger set, without regard to the order of selection. The combination formula is denoted as \( \binom{n}{r} \), often referred to as "n choose r." It calculates the number of combinations of \( n \) items taken \( r \) at a time and is given by:

• \( \binom{n}{r} = \frac{n!}{r! (n-r)!} \)

In our exercise, selecting 4 novels out of 6 available uses the combination formula \( \binom{6}{4} \). This calculation results in 15 unique ways to select the novels.

Similarly, choosing 1 dictionary from 3 available dictionaries uses \( \binom{3}{1} \), resulting in 3 ways to choose the dictionary. The combination formula helps determine these possibilities efficiently without listing them out.

By understanding and applying the combination formula, you can solve many problems involving selections from larger sets, an essential skill in combinatorics and probability.

Problem-Solving Steps in Mathematics

Solving problems in mathematics often requires a structured approach. Following specific problem-solving steps can enhance comprehension and lead to the correct solution of complex problems.

Here's how these steps were applied in the original exercise:

• **Step 1**: Identify the Task - Clearly understand what needs to be solved. In our case, selecting and arranging books with specific conditions.
• **Step 2**: Formulate the Approach - Determine which mathematical concepts or formulas are needed. Here, we used the combination formula to choose novels and dictionaries.
• **Step 3**: Perform Calculations - Execute the necessary calculations, such as factorials and combinations, to find solutions. Factorial calculation helped in arranging novels.
• **Step 4**: Complete and Verify - Multiply or combine results to finalize the solution. Ensure that the solution meets the requirements of the problem, as was done by multiplying the results of selections and arrangements.
• **Step 5**: Interpretation - Based on calculations, interpret the result to select the right answer category.

By adhering to these steps, solving complex mathematical problems becomes more manageable, leading to correct and validated conclusions.