Question

How many 5-digit telephone numbers can be constructed using the digits 0 to 9, if each number starts with 67 and no digit appears more than once?

• A. \(336\)
• B. \(337\)
• C. \(335\)
• D. None of these

Hint:

When repetition is not allowed and order matters, always use permutations: [ ^nPr = fracn!(n-r)! ]

Answer 1

The Correct Option is A

Step 1: Understand the given conditions.

The number is a 5-digit telephone number.
It must start with the digits \(6\) and \(7\).
No digit is repeated.
Step 2: Fix the starting digits. The first two digits are fixed as: \[ 67 \] So, we now need to fill the remaining 3 digits. Step 3: Count the remaining available digits. Digits available: \(0\) to \(9\) \(\Rightarrow 10\) digits Digits already used: \(6, 7\) \[ \text{Remaining digits} = 10 - 2 = 8 \] Step 4: Arrange the remaining digits. The remaining 3 positions can be filled using permutations of 8 digits taken 3 at a time: \[ {}^{8}P_{3} = 8 \times 7 \times 6 = 336 \] Step 5: Final conclusion. The total number of such telephone numbers is \(\boxed{336}\).