Question

How many even numbers are there with three digits such that if $5$ is one of the digits, then $7$ is the next digit?

Answer 1

We have to determine the total number of even numbers formed by using the given condition. So, at units place we can use one of the digits $0,2,4,6,8$. If $5$ is at ten’s place then, as per the given condition, $7$ should be at unit’s place. In such a case the number will not be an even number. So, $5$ cannot be at ten’s and one’s places. Hence, $5$ can be only at hundred’s place. Now two cases arise.

Case $I:$When $5$ is at hundred’s place. If $5$ is a hundred’s place, then $7$ will be at ten’s place. So, unit’s place can be filled in $5$ ways by using the digits $0,2,4,6,8$. So, total number of even numbers $=1\times 1\times 5=5$.

Case $II:$When $5$ in not at hundred’s place. Now, hundred’s place can be filled in $8$ ways ($0$ and $5$ cannot be used at hundred’s place). In ten’s place we can use any one of the digits except $5$. So, ten’s place can be filled in $9$ ways. At unit’s place we have to use one of the even digits $0,2,4,6,8$. So, units place can be filled in $5$ ways. So, total number of even numbers $=8\times 9\times 5=360.$

Hence, the total number of required even numbers $=360+5=365$.