Question

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

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

Answer 1

The correct option is A $216$

We just have to choose 5 digits from given 6 digits
Lets first exclude:
$0=>1,2,3,4,5$ sum $=15$
It will be divisible by $3$
No. of ways $=5!=120$

$1=>0,2,3,4,5$ sum $=14$
It can't be divisible by 3

$2=>0,1,3,4,5$ sum $=13$
It can't be divisible by $3$

$3=>0,1,2,4,5$ sum $=12$
It is divisible by $3$
No. of ways $=5!$
But there will be $4!$ ways in which $0$= 24$waysSo,total$= 5!-4! = 120-24 = 96$

$4=>0,1,2,3,5$ sum $=11$
It can't be divisible by $3$

$5=>0,1,2,3,4$ sum $=10$
It can't be divisible by $3$

So total $=120+96=216$

Hence, option 'A' is correct.