Problem 29 - Algebra for JEE ADVANCED

Question

The number of ways of arranging \(m\) positive and \(n(

Step-by-Step Solution

Answer

The correct answer is \(^{m+1}C_n\).

Step 1: Understand the Problem

We need to arrange \(m\) positive signs and \(n\) negative signs in a row such that no two negative signs are adjacent to each other. This condition needs careful placement of the negative signs among the positive ones.

Step 2: Positioning Positive Signs

Let's first arrange the \(m\) positive signs. These signs create "gaps" where negatives can be placed. There are \(m+1\) gaps: one before the first positive sign, \(m-1\) gaps between any two positive signs, and one after the last positive sign.

Step 3: Placing Negative Signs in Gaps

We need to select \(n\) gaps from the \(m+1\) available positions to place the \(n\) negative signs. This selection ensures no two negative signs are adjacent.

Step 4: Applying Combination Formula

The number of ways to select \(n\) gaps out of \(m+1\) is computed using the combination formula \(^{m+1}C_n\). This gives us the total number of ways to arrange \(n\) negative signs among \(m\) positive signs so that no two negative signs are together.

Key Concepts

Arrangement Problems

Arrangement problems in combinatorics involve organizing a set of elements in a specific order. When tackling these problems, it's essential to consider any specific conditions or restrictions. For example, the original exercise involves arranging positive and negative signs, with the requirement that no two negative signs are adjacent. Such conditions add complexity, turning a simple permutation into a more intricate problem.
Think of arrangement problems like organizing a shelf of books where certain types of books can't be placed next to each other. The challenge is to find all possible arrangements that satisfy these conditions.
In the context of this exercise, you have to creatively use the available space (or gaps) created by positioning positive signs first, before distributing the negative signs. This approach ensures that you adhere to the rule of separation for the negative signs.

Combination Formula

The combination formula is a key concept in combinatorics, used to determine the number of ways to select items from a larger pool. It is represented as \(^nC_k\), which calculates how many different ways you can choose \(k\) items from a set of \(n\) items.
This formula is calculated using the expression: \[ ^nC_k = \frac{n!}{k!(n-k)!} \]Where \(n!\) (n factorial) is the product of all positive integers up to \(n\).
In the original problem, you use this formula to select gaps for placing negative signs. Since there are \(m+1\) gaps created by the positive signs, and you need to place \(n\) negative signs, you calculate the number of ways to choose \(n\) gaps from \(m+1\). This guarantees that the arrangement respects the requirement that no two negatives are adjacent.

Gap Method in Arrangements

The gap method is a technique used to solve arrangement problems, especially when dealing with restrictions like no two similar elements being adjacent. The method involves creating slots or gaps between elements placed first—in this case, the positive signs.
When arranging \(m\) positive signs, they naturally form \(m+1\) gaps around and between them. These gaps become potential locations for placing \(n\) negative signs.
By ensuring that only one negative sign is placed per gap, the gap method satisfies the requirement that no two negative signs are adjacent. This method is quite strategic as it simplifies the placement process by transforming a complex problem into a more straightforward selection issue. Consequently, once gaps are identified, the next step is to apply the combination formula to determine the optimal arrangement.