Question

How many 6 digit telephone numbers can be constructed with the digits 0 to 9 ,if each number starts with 23 and if repetition of digits is not allowed.

• A. 720
• B. 1680
• C. 5040
• D. 10000

Answer 1

The correct option is B

1680

Of the 6 digits, first 2 digits are taken as 2 and 3. The remaining 4 digits are to be filled.

As the repetition is not allowed, and as the 2 digits 2 and 3 are already used, the $3^{rd}$ digit can be filed with 0 or 1 or 4 or 5 or 6 or 7 or 8 or 9 i.e., in 8 ways.

The $4^{th}$ digit can be filled in 7 ways (excluding the numbers in $1^{st}$ three digits)

The $5^{th}$ digit can be filled in 6 ways.

The $6^{th}$ digit can be filled in 5 ways.

So, the number of 5 digit telephone numbers possible = $8\times 7\times 6\times 5$ = 1680

Alternate way:

Of the 6 digits, first 2 digits are taken as 2 and 3. The remaining 4 digits are to be filled.

As the repetition is not allowed, and as the 2 digits 2 and 3 are already used, available digits are 8.

So, the number of ways of filling 4 places with 8 numbers when repetition is not allowed is ${}^8{P}_4$ = 1680