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.
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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: .
The other 3 digits can be: .
. . Each has permutations.
Hence, there are: four-digit numbers that begin with 5.
The number begins with 6: .
The other 3 digits can be: .
. . Each has permutations.
Hence, there are: four-digit numbers that begin with 6.
There are: four-digit numbers.
Five-digit numbers
The digits add up to 18, a multiple of 3.
Hence, any five-digit number will be divisible by 3.
There are choices for the first digit.
The other 4 digits can be permuted in ways.
Hence, there are: five-digit numbers.
Therefore, there are : such numbers.