How many distinct four-digit numbers can be formed by the digits {1, 2, 3, 4, 5, 5, 6, 6}?

Question

Suraj Pandey · Accepted Answer

Understand the problemWe need to form four-digit numbers using the digits {1, 2, 3, 4, 5, 5, 6, 6}.Digits can be repeated as given in the set, but the numbers must be distinct.
Calculate the number of permutationsTotal digits: 8We need to choose 4 out of these 8 digits.
Consider repetitionsFor each selection of 4 digits, we need to consider different cases for repetitions:  1. All digits are unique  2. One digit is repeated
Case 1: All digits are uniqueSelect 4 digits from {1, 2, 3, 4, 5, 6}Number of ways to select 4 out of 6 = C(6, 4) = 15Number of permutations of 4 unique digits = 4! = 24Total = 15 * 24 = 360
Case 2: One digit is repeatedChoose 3 out of {1, 2, 3, 4, 5, 6}Number of ways to select 3 out of 6 = C(6, 3) = 20Number of permutations with one repetition = 4!/2! = 12Total = 20 * 12 = 240
Sum the casesTotal distinct four-digit numbers = 360 + 240 = 600