Repeated Digits in 9-Digit Numbers | JEE Main Permutation Combination Question

Let S = {1, 2, 3, 4, 5, 6, 7, 8, 9}. Let x be the number of 9-digit numbers formed using the digits of the set S such that only one digit is repeated and it is repeated exactly twice. Let y be the number of 9-digit numbers formed using the digits of the set S such that only two digits are repeated and each of these is repeated exactly twice. Then,

Options:

1. 45x = 7y

2. 56x = 9y

3. 21x = 4y

4. 29x = 5y

Correct Answer: 21x = 4y

Explanation

Step 1:

To find x, choose one digit to be repeated twice from 9 digits and choose remaining 7 distinct digits from the remaining 8 digits.

x = 9 × C(8,7) × 9! / 2!

Step 2:

To find y, choose two digits to be repeated twice from 9 digits and choose remaining 5 distinct digits from the remaining 7 digits.

y = C(9,2) × C(7,5) × 9! / (2! × 2!)

Step 3:

Simplifying the ratio of x and y, we obtain:

21x = 4y