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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

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B

1680

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C

5040

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D

10000

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Solution

## 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 3rd 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 4th digit can be filled in 7 ways (excluding the numbers in 1st three digits) The 5th digit can be filled in 6 ways. The 6th digit can be filled in 5 ways. So, the number of 5 digit telephone numbers possible = 8×7×6×5 = 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 8P4 = 1680

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