How many five-digit numbers can be made having exactly two identical digits?

Explanation

Case I : Two identical digits are 0,0.
The number of ways to select three more digits is `""^(9)C_(3)`. The number of arrangements of these five digits is `5!//2!-4! =36`.
Hence, the number of such numbers is
`""^(9)C_(3)xx36=3024`
Case II : Two identical digits are (1,1) or (2,2) or .. or (9,9).
If 0 is included, then number of ways of selection of two more digits is `""^(8)C_(2)`. The number of ways of arrangements of these five digits is `5!//2!-4!2! =48`. Therefore, number of such numbers is `""^(8)C_(3)`. Therefore, number of such numbers is `""^(8)C_(3)xx5!//2! = ""^(8)C_(3)xx60`. Hence, total number of five-digit numbers with identical digits (1,1) or (2,2) or ..(9,9) is
`9xx(""^(8)C_(2)xx48+ ""^(8)C_(3)xx60)=42336`
From (1) and (2), the required number of numbers is
3024+42336=45360.