Question

The number of $5$-digit numbers which are divisible by $4$ and sum of digits is odd, with the digits from the set $\{1,2,3,4,5,6\}$ and repetition of digits is not allowed, is

Answer 1

Since the given number is divisible by $4,$
therefore, the last two digits of the number must be $12,16,32,32,36,52,56,24$ or $64.$
Also, the sum of digits of the required number is an odd number. So, three of its digit must be odd and rest two even.

If the number ends in $12,16,32,32,36,52$ or $56,$ then out of remaining three digits $2$ will be odd and one will be even.
This can be done in number of ways $={}^2{C}_2\times {}^2{C}_1\times 3!\times 6=72$

If the number ends in $24$ or $64,$ then remaining three digits will be odd.
This can be done in number of ways $={}^3{C}_3\times 3!\times 2=12$

$\therefore$ Total number of numbers will be $84$