Question

The number of 3-digit numbers, formed using the digits 2, 3, 4, 5 and 7, when the repetition of digits is not allowed, and which are not divisible by 3, is equal to ______.

Answer 1

Correct Answer: 36

The given digits are:
\[ \{2, 3, 4, 5, 7\}. \]
A number is divisible by 3 if the sum of its digits is divisible by 3. Identify all cases where the sum of three digits is divisible by 3.
The total number of 3-digit permutations is:
\[ P(5, 3) = 5 \cdot 4 \cdot 3 = 60. \]
Now exclude numbers that are divisible by 3. Compute sums of digits for all groups of three: For digits \( (2, 3, 4), (3, 5, 7) \), etc., find cases where sums like \(2 + 3 + 4 = 9\) (divisible by 3).
Count the total valid cases:
Divisible cases: \(6\) (from permutations of divisible groups).
The remaining numbers are:\[ 60 - 24 = 36. \]