Question

Number of $4$ digit numbers that can be formed using the digits $0,1,2,3,4,5$ which are divisible by $6$ when repetition of digits is not allowed are

• A. $24$
• B. $52$
• C. $72$
• D. $48$

Answer 1

The correct option is B $52$

For any number to be divisible by $6$ the number should be even and divisible by $3.$
Now, for number to be even the last digit should be $0,2,4.$

Now, if the last digit is $0$ then first $3$ digits can be $(1,2,3)$ or $(2,3,4)$ or $(3,4,5)$ or $(1,3,5).$
So, possible numbers are $=4\times 3!=24$

Now, if the last digit is $2$ then first $3$ digits can be $(0,1,3)$ or $(0,3,4)$ or $(1,4,5).$
So, possible numbers are $=4+4+3!=14$

Now, if the last digit is $4$ then first $3$ digits can be $(0,2,3)$ or $(0,3,5)$ or $(1,2,5).$
So, possible numbers are $=4+4+3!=14$

Hence, total numbers are $14+14+24=52$