Question

A five digit number is formed by the digits $1,2,3,4,5$ without repetition. The probability that the number formed is divisible by $4$ is $\frac{k}{25}$.Find the value of $k$

Answer 1

Total 5 digit numbers that can be formed without repetition$=5!=120$
For a number to be divisible by 4,the last 2 digits of that number must be divisible by 4.
Hence the unit digit can only be 2 or 4.
When the unit digit is 4,the tens digit can only be 2 $\Rightarrow$ rest of the three digits can be arranged in $3!=6$ ways.
when the unit digit is 2,then the tens digit can be 1 $\Rightarrow$ rest of the three digits can be arranged in $3!=6$ ways.
when the unit digit is 2,then the tens digit can be 3 $\Rightarrow$ rest of the three digits can be arranged in $3!=6$ ways.
when the unit digit is 2,then the tens digit can be 5 $\Rightarrow$ rest of the three digits can be arranged in $3!=6$ ways.
Hence, total possibilities$=6+6+6+6=24$.
Probability$=\frac{24}{120}=\frac{1}{5}=\frac{5}{25}\Rightarrow k=5$