Question

A $5$ digit number divisible by $3$ is to be formed using the numbers $0,1,2,3,4\text{\&}5$ without repetition. The total number of ways this can be done is

• A. $216$
• B. $240$
• C. $600$
• D. $3125$

Answer 1

Answer: (A)

$216$

A number is divisible by $3$ only if the sum of its digits are also divisible by $3$. Here we are supposed to form a $5$ digit number which is divisible by three, if one observes carefully then there are only two combination of $5$ digits from the given pool for which the sum is divisible by $3$ viz. $1,2,3,4\text{\&}5$ and $0,1,2,4\text{\&}5$. So we just have to find the number of ways in which we can form a $5$ digit number using these two pools of digits.
For the first pool, the number of ways $=5!$
For the second pool of digits, number of ways $=4\times 4!$ since zero cannot be placed at the first position and $\therefore$ there are only 4 possibilities for the first place in the number.
Thus, the final answer is $=5!+4\times 4!=120+96=216$