# Question

Phone numbers in the United States consist of 10 digits, a three- digit area code followed by a seven- digit number. Suppose there are several constraints that must be satisfied for a phone number to be valid, such as:

(i) Neither the first digit of any area code nor the first digit of a phone number can be 0, (since 0 is reserved to connect to an operator).

(ii) The second digit of any valid area code must be either 0 or 1.

(iii) The second digit of any valid phone number must not be either 0 or 1 (the second and third constraints are no longer in place but were once used so that the phone company could easily determine whether you were making a local or long distance call).

(iv)The second and third digits of any valid area code cannot both be 1s. (v) The second and third digits of any valid phone number cannot both be 1s. (Three- digit numbers ending in 11 are reserved for special purposes, e. g., emergency, 911, or information, 411).

(a) Given the five constraints listed above, how many valid three- digit area codes are there?

(b) Given the five constraints listed above, how many valid seven- digit phone numbers are there?

(c) How many different 10- digit phone numbers can be constructed under these constraints?

(i) Neither the first digit of any area code nor the first digit of a phone number can be 0, (since 0 is reserved to connect to an operator).

(ii) The second digit of any valid area code must be either 0 or 1.

(iii) The second digit of any valid phone number must not be either 0 or 1 (the second and third constraints are no longer in place but were once used so that the phone company could easily determine whether you were making a local or long distance call).

(iv)The second and third digits of any valid area code cannot both be 1s. (v) The second and third digits of any valid phone number cannot both be 1s. (Three- digit numbers ending in 11 are reserved for special purposes, e. g., emergency, 911, or information, 411).

(a) Given the five constraints listed above, how many valid three- digit area codes are there?

(b) Given the five constraints listed above, how many valid seven- digit phone numbers are there?

(c) How many different 10- digit phone numbers can be constructed under these constraints?

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