Question

A five-digit number divisible by $3$ is to be formed using the digits $0,1,2,3,4\mathrm{and}5$, without repetition. The total number of ways this can be done is _____

Answer 1

Let's calculate the total number of ways by which the number can be formed:

We know that the divisibility rule for $3$ states that a number is completely divisible by $3$ if the sum of its digits is divisible by $3$.

We are given with $6$ digits: $0,1,2,3,4and5$

We need only $5$ digits to form the number

Case I: Using digits $0,1,2,3,4,5;$ the number of ways $=4\times 4\times 3\times 2\times 1=96$

Case II: Using digits $0,1,2,3,4,5;$ the number of ways $=5\times 4\times 3\times 2\times 1=120$

Therefore, the total number formed $=120+96=216$

Therefore, the total number of ways by which the number can be formed $=216$