The total number of 5-digit numbers, formed by using the digits 1, 2, 3, 5, 6, 7 without repetition, which are multiples of 6, is:
Solutions · 1
Official solution
For divisibility by 3, the 5 chosen digits (out of the 6 available) must have a digit sum divisible by 3. The total sum of all 6 digits is 1+2+3+5+6+7=24. Removing one digit leaves a sum divisible by 3 only if the removed digit itself is divisible by 3, i.e. remove 3 or remove 6.
Remove 3 (use 1,2,5,6,7): number of even numbers (ending in 2 or 6) =4×3×2×1×2=48
Remove 6 (use 1,2,3,5,7): number of even numbers (ending in 2 only) =4×3×2×1×1=24
\[\text{Total} = 48+24 = 72\]