# probability

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• Sep 9th 2013, 04:34 AM
Trefoil2727
probability
Find how many distinct number greater than 5000 and divisible by 3 can be formed from digits 3,4,5,6,and 0. Each digit being used at most once in any number.
• Sep 9th 2013, 05:39 PM
Soroban
Re: probability
Hello, Trefoil2727!

Quote:

Find how many distinct numbers greater than 5000 and divisible by 3 can be formed
from digits 3,4,5,6,and 0, each digit being used at most once in any number.

A number is divisible by 3 if its sum of digits is divisible by 3.

Four-digit numbers

The number begins with 5: . $5\,\_\,\_\,\_$
The other 3 digits can be: . $\{0,3,4\},\:\{0,4,6\},\:\{3,4,6\}$
. . Each has $3!$ permutations.
Hence, there are: $3\cdot3! \,=\,18$ four-digit numbers that begin with 5.

The number begins with 6: . $6\,\_\,\_\,\_$
The other 3 digits can be: . $\{0,4,5\},\:\{3,4,5\}$
. . Each has $3!$ permutations.
Hence, there are: $2\cdot3! \,=\,12$ four-digit numbers that begin with 6.

There are: $18 + 12 \,=\,{\color{blue}30}$ four-digit numbers.

Five-digit numbers

The digits $\{0,3,4,5,6\}$ add up to 18, a multiple of 3.
Hence, any five-digit number will be divisible by 3.

There are $4$ choices for the first digit.
The other 4 digits can be permuted in $4!$ ways.
Hence, there are: $4\cdot4! \,=\,{\color{blue}96}$ five-digit numbers.

Therefore, there are : $30 + 96 \,=\,{\color{blue}126}$ such numbers.