UPSC CSE Prelims 2021
The given digits are 2, 2, 3, 3, 3, totaling five digits. The objective is to form 5-digit numbers greater than 30000 using all these digits.
For a 5-digit number to be greater than 30000, its first digit must be 3. If the first digit were 2, the resulting number would be less than 30000.
By fixing 3 as the first digit, the remaining digits available for arrangement are 2, 2, 3, 3. These are four digits, with the digit 2 appearing twice and the digit 3 appearing twice.
The number of distinct permutations of these four digits (n=4) with repetitions (p₁=2 for digit 2, p₂=2 for digit 3) is calculated using the formula:
Number of ways = n! / (p₁! p₂!)
Substituting the values:
Number of ways = 4! / (2!2!) = (4 × 3 × 2 × 1) / ((2 × 1) × (2 × 1)) = 24 / 4 = 6.
Therefore, 6 distinct numbers greater than 30000 can be formed using the given digits. These numbers are: 32233, 32323, 32332, 33223, 33232, 33322.
Options 1 (3), 3 (9), and 4 (12) are incorrect as they do not correspond to the calculated number of distinct 5-digit numbers greater than 30000 that can be formed using the specified set of digits, based on the principles of permutations with repetitions.