Question

Given the digits 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, how many four-digit numbers can be formed ?

Answer 1

Such four-digit numbers can

(a) Contain all different digits, any four chosen from 1, 2, 3, 4 and 5 and arranged to form four digit number and this is done in ⁵P₄ = 120 ways.

(b) Contain one repeated pair, the other two different, numbers like 1123, 3345, 3435, … and this is done in  ⁵C₁ x  ⁴C₂  x 4!/2! = 360 ways.

(c) Contain two repeat pairs, numbers like 1122, 1212, 3113, …, and this is done in  ⁵C₂ 4!/2!2! = 60 ways.

Therefore, total number of numbers = 540.