Question

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?

Answer 1

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

3

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) 

P(n, r) = \(\frac{n!}{(n-r)!}\)

∴ 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.