Question

The number of five-digit positive integers divisible by $3$ that can be formed using the digits $0,1,2,3,4,5$ without any repetition is

• A. $120$
• B. $96$
• C. $216$
• D. $625$

Answer 1

The correct option is C $216$

A number is divisible by $3$ if the sum of its digits is divisible by $3.$
We observe that $1+2+3+4+5=15$ is divisible by $3.$
Let the number be $d_1{d}_2{d}_3{d}_4{d}_5.$

If $0$ is not included then
The number of ways $=5!=120$

If $0$ is included, then $3$ has to be removed for divisibility by $3.$
Case $1:$ If $d_1=1$
Number of ways $=4!$

Case $2:$ If $d_1=2$
Number of ways $=4!$

Case $3:$ If $d_1=4$
Number of ways $=4!$

Case $4:$ If $d_1=5$
Number of ways $=4!$

Total number of ways $=120+4\times 24=216$