Question

A five digit number divisible by $3$ is to be formed by using the numerals $0$, $1$, $2$, $3$, $4$ and $5$ without repetition. The total number of ways in which can be done is

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

Answer 1

Answer: (A)

$216$

Answer 2

Since, a five-digit number is formed using the digits \\{0,1,2,3,4 and 5\\} divisible by $3$ i.e. only possible when sum of the digits is multiple of three.
Case I Using digits $0, 1, 2, 4, 5$
Number of ways $= 4 \times 4 \times 3 \times 2 \times 1 = 96$
Case II Using digits $1, 2, 3, 4, 5$
Number of ways $= 5 \times 4 \times 3 \times 2 \times 1 = 120$
$\therefore$ Total numbers formed $= 120 + 96 =216$