Question

Find the number of three digit numbers that are divisible by $2$ but not divisible by $6$ and are formed with the digits $1,2,3,4,6$ when repetition is not allowed

• A. $10$
• B. $12$
• C. $20$
• D. $25$

Answer 1

Answer: (C)

$20$

A number is divisible by $2$ if the last digit is an even number

A number is divisible by $6$ if it is divisible by both $2$ and $3$

So, a three digit number formed $1,2,3,4,6$ is divisible by $2$ if the last digits are $2,4$ or $6$

And it is not divisible by $6$ if it is not divisible by $3$

So, we need to find numbers divisible by $2$ and not by $3$ with no repetition

Case 1 :

$(1,4)(3,6)$

$\Rightarrow$Number of possiblities $=(2!)2$

Case 2 :

$(1,2)(1,3)(1,6)(3,6)$

Number of Possiblities $=(2!)4$

Case 3 :

$(1,3)(1,4)(2,3)(3,4)$

Number of Possiblities $=(2!)4$

Hence the correct answer is $20$