Question

The number of $4$-digit numbers that can be made with the digits $1,2,3,4$ and $5$ in which at least two digits are identical, is

• A. 505
• B. $4^5-5!$
• C. 600
• D. None of these

Answer 1

Answer: (A)

First we'll calculate the number of ALL $4$-digit numbers that can be made with them. Then we'll calculate and subtract the number of $4$-digit numbers that don't have any of the digits identical.
To find all possible $4$-digit numbers that can be made.
1. There are $5^4$ ways $=625$ ways to choose the $1st4$ digits, which is all of them.
Now we must subtract the number of ways in which there are no two digits alike $=5!$ ways $=120$ ways to choose the $1st4$ digits, which is all of them.
So $\Rightarrow =625-120=505$