Question

A five-digit number is formed using the digits 1, 2, 3, 4, and 5 without repetition. What is the probability that the number is divisible by 4?

• A. \(\frac{1}{5}\)
• B. \(\frac{5}{6}\)
• C. \(\frac{4}{5}\)
• D. None of these

Hint:

Only the last two digits decide if a number is divisible by 4, so check which digit pairs work.

Answer 1

The Correct Option is A

There's a shorter way to get here, using symmetry instead of counting all 120 arrangements directly.

Whether the whole number is divisible by 4 depends only on its last two digits, not on how the first three digits are arranged. So we can just look at the last two positions and ignore the rest.

Pick 2 digits out of the 5 to sit in the last two places, in order. The number of ways to do this is $5 \times 4 = 20$, since there are 5 choices for the second-last digit and 4 remaining choices for the last digit.

Out of these 20 ordered pairs, only 4 give a number divisible by 4: 12, 24, 32, 52, checked by dividing each by 4.

Because the leftover 3 digits can be arranged in any order regardless of which pair sits at the end, that factor of $3!$ appears in both the favourable count and the total count, so it cancels out. We are left with $\frac{4}{20} = \frac{1}{5}$.

Let's summarize:

• Only the last two digits decide divisibility by 4
• Favourable last-two-digit pairs: 4 out of 20 possible ordered pairs
• Probability: $\frac{4}{20} = \frac{1}{5}$

This matches option A.