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  4. 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

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Q. 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

BITSATBITSAT 2019

Solution:

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 $1st 4$ 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 $1st 4$ digits, which is all of them.
So $\Rightarrow =625-120=505$