Problem 69 - A Complete Resource Book in Mathematics for JEE Main

Question

The number of different 7 digit numbers that can be written using only the three digits 1,2 and 3 with the condition that the digit 2 occurs twice in each number is (A) \({ }^{7} P_{2} 2^{2}\) (B) \({ }^{7} C_{2} 2^{5}\) (C) \({ }^{7} C_{2} 5^{5}\) (D) None of these

Step-by-Step Solution

Answer

Option (B) \( ^7C_2 \times 2^5 = 672 \).

Step 1: Identify Placement for Digit 2

To solve for the number of 7-digit numbers using the digits 1, 2, and 3 such that the digit 2 appears exactly twice, we first need to select the places for the two occurrences of digit 2. As there are 7 positions available, we use combinations to determine in how many ways we can choose 2 positions from 7.The formula for combinations is given by \( ^nC_k = \frac{n!}{k!(n-k)!} \). Applying this, we have:\[ ^7C_2 = \frac{7!}{2!(7-2)!} = \frac{7 \times 6}{2 \times 1} = 21 \]

Step 2: Distribute Remaining Digits

After placing the two digit 2s, there are 5 positions left. These positions can only be filled with the digits 1 and 3, and each digit can be repeated any number of times. Therefore, for each of these 5 positions, we have 2 choices (1 or 3).So, the total number of ways to fill the remaining 5 positions is \( 2^5 \). Calculating this gives:\[ 2^5 = 32 \]

Step 3: Calculate the Total Number of Combinations

Finally, to find the total number of valid 7-digit numbers, we multiply the number of ways to place the two 2s by the number of ways to fill the remaining positions:\[ ^7C_2 \times 2^5 = 21 \times 32 = 672 \]

Step 4: Find the Correct Answer Option

The calculated total corresponds to option (B). The expression for our answer is given as \( ^7C_2 \times 2^5 = 672 \), which matches with: (B) \( ^7C_2 2^5 \). Therefore, option (B) is the correct answer.

Key Concepts

Permutation and Combination

In combinatorics, permutation and combination are essential concepts used to solve problems involving arrangements and selections. Understanding the difference between these two is fundamental in discrete mathematics.
A **permutation** considers the order of selection. Therefore, it is used when the arrangement or sequence of the elements is essential. For example, the arrangements of letters in a word require a permutation.
In contrast, a **combination** does not account for the order and is used when the selection is made from a larger group without regard to the sequence. This is aptly demonstrated in the given exercise, where we needed to select positions for the digit '2' among the available 7 positions. We used the combination formula to calculate this:

• The formula for finding combinations is: \[ ^nC_k = \frac{n!}{k!(n-k)!} \]
• This indicates choosing 'k' positions from 'n'.

By applying the combination formula, we find it is possible to choose the two '2' positions out of 7 available spots in 21 different ways.

Binary Choice Problems

Binary choice problems involve decisions that have exactly two possible options. Often termed as 'either/or' decisions, these problems frequently appear in digital systems and coding.
In our exercise, this concept is at work when distributing remaining slots after placing the digit 2. The leftover positions could be filled using either the digit '1' or '3'. This setup illustrates the core of binary choice problems: deciding between two distinct options for each choice.
Since we have 5 remaining positions, each with 2 choices (1 or 3), we calculate:

• The total combinations using: \[ 2^5 \]
• This computation arises because each of the 5 positions has 2 possibilities, leading to 32 total combinations.

This classical example of binary choice highlights how discrete scenarios are best solved by counting the number of effective options repeatedly available at each decision point.

Discrete Mathematics

Discrete mathematics focuses on studying mathematical structures that are fundamentally discrete rather than continuous. This field deals primarily with countable, distinct elements, which means it is dedicated to topics where values can only be certain distinct numbers.
Problems in discrete mathematics often involve understanding sequences, graphs, or logical statements. The solution to exercises of this nature typically focuses on counting, as seen in our example problem. Here, the task was to count the distinct ways to construct a specific arrangement of numbers using particular digits.
**Key Principles of Discrete Mathematics:**

• Counting - as was applied to first select positions for '2's and then choices for remaining numbers.
• Combining choices through multiplication, to derive total possibilities from multiple independent decisions.

Thus, discrete mathematics provides essential tools for a clear understanding of problems that deal with distinct, countable configurations, whether involving arrangements, combinations, or permutations.