How many distinct 8-digit numbers can be formed by rearranging the digits of the number 11223344 such that odd digits occupy odd positions and even digits occupy even positions ?

(a) 12
(b) 18
(c) 36 ✓ UPSC's answer
(d) 72

Why the answer is (c)

• The number 11223344 contains two 1s, two 2s, two 3s, and two 4s.

• Odd digits are 1, 1, 3, 3 (four digits) and even digits are 2, 2, 4, 4 (four digits).

• There are 4 odd positions (1st, 3rd, 5th, 7th) and 4 even positions (2nd, 4th, 6th, 8th).

• The number of ways to arrange the four odd digits (1, 1, 3, 3) in the four odd positions is 4! / (2! * 2!) = 24 / 4 = 6.

• The number of ways to arrange the four even digits (2, 2, 4, 4) in the four even positions is 4! / (2! * 2!) = 24 / 4 = 6.

• The total number of distinct 8-digit numbers is the product of these arrangements: 6 * 6 = 36.

Why the other options are wrong

(a) 12: 12 is incorrect because it likely results from miscalculating the permutations of the odd or even digits, such as treating them as distinct or using incorrect factorial division.
(b) 18: 18 is incorrect because it does not correspond to the product of the valid permutations of the odd and even digit sets.
(d) 72: 72 is incorrect because it assumes all digits are distinct or fails to account for the repetition of digits in the permutation calculation.

Asked in the CSAT Paper II of the UPSC Civil Services Preliminary Examination 2023, held on 28 May 2023. Question and answer key: Union Public Service Commission. Explanation: UPSC Answer Check.