Question

How many $4$-digit numbers can be formed from digit $1,1,2,2,3,3,4,4,5,5$?

Answer 1

Case (1) ,

All the $4$ digits are different in the number.

Then the digits can be arranged in ${}^5{\text{P}}_4=\frac{5!}{(5-4)!}=120$ ways to form the number.

Case (2)

When one digit is repeated out of $1,2,3,4,5$ and other two digits used are different.

There are $5$ ways to choose which digit is repeated.

There are ${}^4{\text{C}}_2=6$ ways to choose the non repeated digits.

And $\frac{4!}{\left(2\right)!}=12$ ways to arrange all the digits.

So, with one repetition, total ways $=5\left(6\right)\left(12\right)=360$

Case (3)

When two digits are repeated out of $1,2,3,4,5$

There are ${}^5{\text{C}}_2=10$ ways to choose the repeated digits.

And $\frac{4!}{2!2!}=6$ ways to arrange all the digits.

So, with two repetition, total ways $=10\left(6\right)=60$

So, total 4-digit numbers $=120+360+60=540$