Question

Find the number of $4-$ digit number that can be formed using the digits $1,2,3,4,5$ if no digit is repeated. How many of these will be even?

Answer 1

Finding $4-$ digit numbers with digits $1,2,3,4,5$ if no digit is repeated.
Total no.of digits $=5\Rightarrow n=5$
$4$ digit numbers to be formed $\Rightarrow r=4$
$\therefore$ Required number of $4-$ digit numbers $={}^n{P}_r={}^5{P}_4=\frac{5!}{(5-4)!}=120$

Finding $4-$ digit numbers with digits $1,2,3,4,5$ if no digit is repeated.
The given digits are $1,2,3,4,5$ even digits are $2,4$
For an even number, units place should be even
So, no. of ways to fill units place with $2$ even digits $={}^2{P}_1=2$
No.of different ways to fill remaining $3$ places with other $4$ digits $={}^4{P}_3=24$

$\therefore$ No.of four digit even numbers $=24\times 2=48$.