Step 1: Divisibility Rule for 3
A number is divisible by 3 if the sum of its digits is divisible by 3.
Step 2: Selecting the Digits
The sum of all given digits is:
We need to choose 5 digits out of 6. This is equivalent to excluding one digit. For the sum of the remaining 5 digits to be divisible by 3, the excluded digit must also be a multiple of 3 (since 15 is a multiple of 3).
The digits in the set that are multiples of 3 are 0 and 3.
Case 1: Exclude the digit 0 The set of digits is {1,2,3,4,5}. The number of ways to arrange these 5 digits to form a five-digit number is:
Case 2: Exclude the digit 3 The set of digits is {0,1,2,4,5}. In this case, the first digit cannot be 0.
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Number of choices for the first digit (thousands place): 4 (any of 1,2,4,5)
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Number of ways to arrange the remaining 4 digits in the remaining 4 places: 4! Total ways for Case 2:
Step 3: Total Number of Ways
The total number of five-digit numbers is the sum of both cases:
