Question

Find the number of $4$ -digit numbers 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

$4$ digits can be selected in ${}^5{C}_4$ ways

$4$ digits can be arranged in $4!$ ways

so, the number of 4 digits number formed are ${}^5{C}_4\times 4!=5\times 4!=5!=120$

for even number unit digit of the number has to be $2$ or $4$

unit digit of the number can be selected in ${}^2{C}_1$ ways

number other than unit digit can be formed in ${}^4{C}_3\times 3!$ ways$=4\times 6=24$ ways

total number of even number will be ${}^2{C}_1\times 24=48$