A 5- digit number divisible by 3 is to be formed using the digits… - Vidyadip

MCQ

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

✓
$216$
B
$600$
C
$240$
D
$3125$

✓

Answer

Correct option: A.

$216$

A number is divisible by $3$ when the sum of the digits of the number is divisible by $3$.
Out of the given $6$ digits, there are only two groups consisting of $5$ digits whose sum is divisible by $3$.
$= 1 + 2 + 3 + 4 + 5 = 15$
$= 0 + 1 + 2 + 4 + 5 = 12$
Using the digits $1, 2, 3, 4$ and $5,$ the $5$ digit numbers that can be formed $= 5!$ Similarly, using the digits $0, 1, 2, 4$ and $5,$ the number that can be formed $= 5! - 4! $ since the first digit cannot be $0$
$\therefore$ Total numbers that are possible $= 5! + 5! - 4! = 240 - 24 = 216$

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