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  4. A five digit number divisible by 6 is to be formed by using the digits 0,1,2,3,4 and 8 without repetition. The total number of ways in which this can be done is

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Q. A five digit number divisible by 6 is to be formed by using the digits $0,1,2,3,4$ and 8 without repetition. The total number of ways in which this can be done is

Permutations and Combinations

Solution:

Since $0+1+2+3+4+8=18$, the 5 digit number will be divisible by 3 if either 0 or 3 is not used.
When 0 is not used. For the unit place we have 3 choices (2,4 or 8$)$ and for the remaining place we have 4 ! Choices.
When 3 is not used
In this case, 0 is used at the unit's place, the number of choices is 4 ! If 0 is not used at the unit's place, then unit's place can be filled up in 3 ways and the remaining places in $(4 !-3$ !) ways.
Thus, required number of numbers
$=3(4 !)+4 !+3(4 !-3 !)=150$