Problem 106 - Algebra for JEE ADVANCED

Question

There are \((n+1)\) white and \((n+1)\) black balls each set numbered 1 to \(n+1\). The number of ways in which the balls can be arranged in a row so that the adjacent balls are of different colours is a. \((2 n+2) !\) b. \((2 n+2) ! \times 2\) c. \((n+1) ! \times 2\) d. \(2\\{(n+1) !\\}^{2}\)

Step-by-Step Solution

Answer

The answer is d: \(2{(n+1)!}^2\).

Step 1: Recognize the Arrangement Pattern

For the balls to be arranged in such a way that adjacent ones are of different colors, the arrangement must strictly alternate between white and black. Since there are equal numbers of white and black balls, one feasible arrangement is starting with a white ball and continuing with alternating colors. The other arrangement would be starting with a black ball and alternating.

Step 2: Count Starting with White

Consider the arrangement starting with a white ball. The sequence will be white, black, white, black, and so on. There are ! distinguishable ways to arrange the white balls and another ! distinguishable ways to arrange the black balls in this alternating pattern.

Step 3: Count Starting with Black

Similarly, if we start with a black ball and alternate, the pattern will be black, white, black, white, and so on. As in the previous case, there are ! distinguishable ways to arrange the black balls and ! ways for the white balls.

Step 4: Combine Both Scenarios

Since the sequences are independent, both the white-start pattern and black-start pattern can coexist. Therefore, we add their contributions. Thus, the total number of possible arrangements is ! imes ! (for starting with a white ball) plus ! imes ! (for starting with a black ball).

Step 5: Calculate Total Ways

To finalize, sum the number of arrangements for both cases: Starting with white and starting with black. This results in 2 ! ! .

Key Concepts

Combinatorics

Combinatorics is a branch of mathematics dealing with counting and arranging objects. It helps us find the total number of ways to arrange or combine different items, often using mathematical structures like permutations and combinations. For this exercise, understanding how to alternate black and white balls in a sequence is essential.
Conceptualizing this involves viewing the sequence as a permutation problem with restrictions: black and white must alternate. Combinatorics helps us calculate these permutations. This requires understanding how to arrange the colored balls given that they must not have the same color next to each other. In such problems, it is crucial to consider the constraints as they dramatically reduce the number of valid arrangements, showcasing the practical application of combinatorial techniques in problem-solving.
By embracing these simple counting principles, combinatorics allows us to determine exactly how many valid sequences can be formed, helping to solve problems with specific conditions.

Alternating Sequences

Alternating sequences involve arranging items in a pattern where they follow a specific order or set of rules, such as alternating colors, in this case. The key to solving this alignment of balls lies in understanding the sequence pattern: either start with a white ball and alternate with black, or start with a black ball and alternate with white.
This alternating requirement dictates two different sequences:

• White, Black, White, Black, and so forth.
• Black, White, Black, White, and so on.

In both sequences, each ball must strictly follow the alternating pattern, meaning the first color determines the order of the entire sequence. These constraints create the boundaries for counting valid permutations, ensuring adjacent balls are never the same color. Understanding these rules helps clarify how alternating sequences function, guiding learners to predict and calculate the number of such unique arrangements formed under given conditions.

Factorial Calculations

Factorial calculations play a crucial role in evaluating permutations, especially in combinatorics where arrangements and order are critical. The factorial of a number, denoted as \(n!\), is the product of all positive integers up to that number. In this exercise, factorials are used to count the number of distinguishable ways to arrange the white and black balls.
Given \(n+1\) white and \(n+1\) black balls, factorials calculate every possible arrangement of each colored set independently:

• Arranging \(n+1\) white balls is \( (n+1)! \).
• Arranging \(n+1\) black balls is \( (n+1)! \).

By understanding how factorials compactly represent these arrangements, you can easily assess permutations in complex scenarios with restrictions, like alternating sequences. Factorial calculations thus simplify and systematize the seemingly intricate task of counting large numbers of permutations effortlessly.