# How many 6-digit telephone numbers can be constructed with digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 if each number starts with 35 and no digit appears more than once?

Given, numbers are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9

The total number of these digits are 10

To find: 6-digit telephone numbers starting with 35 and no digit appears more than once

So, fix the position of first two digits as 3 and 5

Now, we need to fill these 4 remaining places

Remaining number of digits are 8 (0, 1, 2, 4, 6, 7, 8, 9)

Arrange these 8 numbers at 4 places.

Formula used:

Number of arrangements of n things taken r at a time = P(n, r)

The total number of ways

= the number of arrangements of 8 things taken 4 at a time

= P(8, 4)

= 8 × 7 × 6 × 5

= 1680

Hence, possible number of 6-digit telephone numbers starting with 35 are 1680

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