Question

Even numbers are formed with three digits such that if $5$ is one of tho digits then $7$ is the next digit. The number of such numbers is (Repetition allowed.)

• A. $360$
• B. $365$
• C. $325$
• D. $300$

Answer 1

Answer: (B)

$365$

Note that of the three digit numbers $5$ is to be followed by $7$ i.e. $5,7$ should appear together. Since the number is to be even, the unit place must be occupied by $0,2,4.6,8$.
Case I. Beginning with $5$

$H$	$T$	$U$
$1$	$1$	$5$
$\left(5\right)$	$\left(7\right)$	$(0,2,4,6,8)$

Any of $0,2,4,6,8$
(5 numbers)
Note 5 can't come in middle place because then the last place has to be seven and it will not be even.
Case II. Not beginning with 5
Case I. Beginning with 5

H	T	U
8	9	5
($\ne 0$,	$\ne 5$)	$\ne 5$

Number of ways $=8\times 9\times 5=360$
Total ways = $360+5=365$