Question

How many $4-$digit number can be from digits $1,1,2,2,3,3,4,4,5,5$

Answer 1

When all the digits are different, four digits can be formed in ${}^5{P}_4$ ways,i.e $\frac{5!}{1!}=120$ways,

With one repetition, the four digits can be formed in

${}^5{C}_1{\times }^4{C}_2\times \frac{4!}{2!}=5\times 6\times 12=360$ways ,

With double repetition, the four digit numbers can be formed in ${}^5{C}_2\times \frac{4!}{2!\times 2!}=60$ways.

So total number of four digit numbers $=120+360+60=540$