Problem 160 - A Complete Resource Book in Mathematics for JEE Main

Question

From 6 different 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

The correct answer is (D) at least 1000.

Step 1: Select the Novels

First, we must choose 4 novels from the 6 available. This can be done in \( \binom{6}{4} \) ways. The formula for calculating combinations is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \). Thus, the number of ways to choose 4 novels from 6 is \( \binom{6}{4} = \frac{6!}{4!2!} = 15 \) ways.

Step 2: Select the Dictionary

We can select 1 dictionary from 3 available dictionaries. This can be done in \( \binom{3}{1} = 3 \) ways.

Step 3: Arrange the Novels

After selecting the novels, we need to arrange them on the shelf. There are 4 novels, so they can be arranged in \( 4! \) ways. Calculating gives \( 4! = 24 \) ways.

Step 4: Position the Dictionary

The dictionary must always be in the middle position of the row consisting of 5 books (4 novels + 1 dictionary). Thus, there is exactly 1 way to place the dictionary in the middle.

Step 5: Calculate Total Arrangements

Multiply the number of ways to select the novels, select the dictionary, arrange the novels, and position the dictionary. The total number of arrangements is given by \( 15 \times 3 \times 24 \times 1 = 1080 \).

Step 6: Conclusion

The total number of arrangements is 1080, which falls into the category "at least 1000." Hence, the correct answer is (D) at least 1000.

Key Concepts

Understanding Permutations

Permutations are all about arranging items in a specific order. In this exercise, permutations help us figure out how to arrange books on a shelf. Imagine you have 4 novels selected. The sequence in which these are placed matters greatly. Because of this, we use permutations. When arranging the novels, we use the factorial function, dedicated to permutations, to figure out all possible orders.

A key thing to remember about permutations is that the arrangement is different every time you swap even two books. If our sequence had been novels A, B, C, and D, a new permutation would be created if we changed this to B, A, C, D. Thus, permutations care a lot about order! Using the formula for permutations of 4 novels, which is given by \( 4! \), delivers 24 unique ways to arrange these 4 novels in order, as shown in the solution of the exercise.

• Use factorials for counting permutations
• Order matters and each arrangement is unique
• Factorials are central in computing permutations

Grasping Combinations

Combinations are different from permutations because the order does not matter. In combinations, selecting items are where you see the big-picture application. In our problem, combinations are useful in choosing which novels to place on the shelf. Here, we sought to choose 4 novels from a selection of 6. Combinations help determine how many sets of 4 novels we could have, regardless of order.

The formula we use is \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \), where \( n \) is the total number of items to choose from, and \( r \) is the number of items we choose. For this exercise, choosing 4 novels out of 6 is calculated through \( \binom{6}{4} = 15 \). This means there are 15 ways to pick 4 novels, independent of arrangement. Remember, combinations emphasize groups, not sequences.

• Order does not matter in combinations
• Useful for choosing groups
• Formula: \( \binom{n}{r} = \frac{n!}{r!(n-r)!} \)

Factorials Made Simple

Factorials are a fundamental building block in both permutations and combinations. A factorial, represented with an exclamation mark (e.g., \( n! \)), is a product of all natural numbers up to a given number. For instance, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorials grow incredibly quickly as numbers increase. They serve to measure large sets of possibilities.

In permutations, factorials help us account for all possible orders of items, while in combinations, factorials help to figure out the number of ways to pick items without considering order. Understanding how to use factorials is crucial for solving a variety of combinatorics problems.

• Factorial notation: \( n! \)
• Used for permutations and combinations
• Helps calculate large numbers of possibilities