Question

Let $m$ (respectively, $n$) be the number of 5-digit integers obtained by using the digits 1,2,3,4,5 with repetitions (respectively, without repetitions) such that the sum of any two adjacent digits is odd. Then $\frac{m}{n}$ is equal to

• A. 9
• B. 12
• C. 15
• D. 18

Answer 1

The correct option is C 15

Given :
The digits are $1,2,3,4,5$
So number of even digits $=2$
So number of odd digits $=3$
when the sum of two adjacent digits are odd, so one is even one is odd.

Case 1 : repetitions is not allowed [$n$]
OEOEO is only arrangement possible (O - odd number, E - evn number)
so $n=3\times 2\times 2\times 1\times 1=12$

Case 2 : repetition is allowed [$m$]
OEOEO
so $m=3\times 2\times 3\times 2\times 3=108$
EOEOE
so $m=2\times 3\times 2\times 3\times 2=72$
total $m=108+72=180$

Hence $\frac{m}{n}=\frac{180}{12}=15$