At a certain university in the U. S., all phone numbers are 7-digits long and start with either 824 or 825.

(a) How many different phone numbers are possible?

(b) How many different phone numbers are there in which the last four digits are all different?

Part A) The total different phone numbers are 20,000

Part B) The total numbers in which the last four digits are all different is 10,080

Step-by-step explanation:

Consider the provided information.

At a certain university in the U. S., all phone numbers are 7-digits long and start with either 824 or 825.

Part (a) How many different phone numbers are possible?

If the numbers are start with 824:

The total digits are 7 out of which 3 are fixed (i. e 824).

For rest of 4 digits we have 10 digits.

10*10*10*10=10,000

If the numbers are start with 825:

The total digits are 7 out of which 3 are fixed (i. e 825).

For rest of 4 digits we have 10 digits.

10*10*10*10=10,000

Hence, the total number of ways are: 10,000+10,000=20,000

(b) How many different phone numbers are there in which the last four digits are all different?

If the numbers are start with 824:

The total digits are 7 out of which 3 are fixed (i. e 824).

Rest 4 digits should be different. That means if we select any number the next number must be other than the selected number and same for the rest of the numbers.

10*9*8*7=5,040

If the numbers are start with 825:

The total digits are 7 out of which 3 are fixed (i. e 825).

Rest 4 digits should be different.

10*9*8*7=5,040

Hence, the total number of ways are: 5,040+5,040=10,080