Question

The number of 5-digit numbers that can be formed using the digits \(1,2,3,4,5,6\) without repetition and divisible by \(5\) is:

• A. \(60\)
• B. \(120\)
• C. \(240\)
• D. \(720\)

Hint:

For divisibility by \(5\), always fix the last digit first and then arrange the remaining digits.

Answer 1

The Correct Option is B

Concept: A number is divisible by \(5\) if its last digit is \(0\) or \(5\). Since only the digit \(5\) is available, every valid number must end with \(5\).

Step 1: Fix the last digit Last digit: \[ 5 \] Now we need to fill the remaining four positions. Available digits: \[ 1,2,3,4,6 \] Total digits available \(=5\).

Step 2: Arrange four digits in four places Number of arrangements: \[ {}^{5}P_{4} = \frac{5!}{(5-4)!} \] \[ = \frac{5!}{1!} \] \[ =5\times4\times3\times2 \] \[ =120 \]

Step 3: Write the result Hence the total number of required numbers is \[ 120 \] Final Answer: \[ \boxed{120} \]