How many positive five-digit multiples of 3 can be formed using the digits 0, 1, 2, 3, 4, and 5, without repeating any digit?

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Correct answer marked below

120

181

216

Correct

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Official Explanation

Official

Official Solution:

First step: Our goal is to determine which 5 digits out of the given 6 will form a 5-digit number divisible by 3.

We have six digits: 0, 1, 2, 3, 4, 5. Their sum is

15

.

For a number to be divisible by 3, the sum of its digits must be divisible by 3.

As the sum of the six given numbers is 15 (divisible by 3), the only 5-digit combinations that would yield a number divisible by 3 are

15-0 = \{1, 2, 3, 4, 5\}

and

15-3 = \{0, 1, 2, 4, 5\}

.

No other combinations of 5 digits from the given 6 will result in a number divisible by 3.

Second step: We have two sets of numbers:

\{1, 2, 3, 4, 5\}

and

\{0, 1, 2, 4, 5\}

.

Let's find out how many 5-digit numbers can be formed using these two sets:…

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What this question tests

What it tests

Whether you can count arrangements and selections correctly — permutations vs. combinations.

Common trap

Overcounting: treating a selection as ordered when order does not matter, or forgetting to divide by duplicates.