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

Suraj Pandey

Understand the problem
We 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 permutations
Total digits: 8
We need to choose 4 out of these 8 digits.

Consider repetitions
For 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 unique
Select 4 digits from {1, 2, 3, 4, 5, 6}
Number of ways to select 4 out of 6 = C(6, 4) = 15
Number of permutations of 4 unique digits = 4! = 24
Total = 15 * 24 = 360

Case 2: One digit is repeated
Choose 3 out of {1, 2, 3, 4, 5, 6}
Number of ways to select 3 out of 6 = C(6, 3) = 20
Number of permutations with one repetition = 4!/2! = 12
Total = 20 * 12 = 240

Sum the cases
Total distinct four-digit numbers = 360 + 240 = 600