Problem 28 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The numbers \(1,2,3, \ldots, n\) are arranged in a random order. The probability that the digits \(1,2,3, \ldots k(k

Step-by-Step Solution

Answer

The correct answer is (D) \(\frac{(n-k+1)!}{n!}\).

Step 1: Understanding the Problem

We need to arrange the numbers \(1, 2, 3, \ldots, n\) randomly such that the sequence \(1, 2, 3, \ldots, k\) appear as neighbors in that exact order.

Step 2: Treating the Sequence as a Single Unit

Since the sequence \(1, 2, 3, \ldots, k\) must appear in that exact order as neighbors, treat them as a single unit or a block. This block now becomes a single element.

Step 3: Counting the Total Number of Arrangements

Originally, there are \(n\) numbers. By treating the sequence as one block, we reduce the problem to placing \((n-k+1)\) elements (the block plus the remaining \(n-k\) numbers). The total number of ways to arrange these \((n-k+1)\) blocks is \((n-k+1)!\).

Step 4: Finding the Probability

The probability of any one specific arrangement happening out of the possible \(n!\) arrangements is given by the ratio \(\frac{\text{favorable outcomes}}{\text{total outcomes}} = \frac{(n-k+1)!}{n!}\).

Step 5: Choosing the Correct Answer

By comparing this calculated probability with the given options, we find that option (D) \(\frac{(n-k+1)!}{n!}\) matches the calculated probability.

Key Concepts

Combinatorics

Combinatorics is the branch of mathematics dealing with counting, combination, and permutation. It's the key to solving many complex problems where you need to arrange or select objects in a specific way. Combinatorics provides us with the tools to compute probabilities by determining how many different ways a set of items can be arranged or grouped.

In this exercise, we use combinatorial principles to count potential arrangements of numbers. The major goal is to understand how many ways we can arrange a sequence, with a particular set of numbers needing to appear next to each other. Thus, combinatorics helps us calculate the distinct orderings and aids in evaluating likelihoods or probabilities associated with events.

Mathematical Probability

Mathematical probability is a field focusing on quantifying the likelihood of events. It provides a numerical measure ranging from 0 to 1, where 0 indicates impossibility and 1 signifies certainty. Understanding probability enables us to evaluate how likely a specific outcome is, given a set of possible outcomes.

In this problem, mathematical probability is at work in calculating the likeliness of a specific arrangement of numbers. By considering all the possible arrangements (total outcomes) and those arrangements where the sequence appears as neighbors (favorable outcomes), we can compute the probability. The formula used here is: \( \frac{\text{favorable outcomes}}{\text{total outcomes}} \). With a total of \( n! \) ways to arrange \( n \) numbers, the favorable arrangements are reduced to \( (n-k+1)! \). This explains why we select the ratio \( \frac{(n-k+1)!}{n!} \), which matches the answer in choice D.

Permutation and Combination

Permutation and combination are essential concepts in combinatorics. Permutations relate to arrangements where order is important, while combinations focus on selections where order doesn't matter. This problem is an example of a permutation since the order in which the numbers appear is critical.

Here, the challenge is arranging the numbers 1 to \( n \) such that a specific sequence appears in the exact order. We treat the sequence as a single unit or block, reducing the task to finding permutations of a smaller set of elements. Only distinct orderings of these blocks are favorable, making the permutation formula \( (n-k+1)! \) crucial in determining the probability. Thus, permutation and combination principles enable us to solve the problem by calculating distinct sequences.