Question

How many phone numbers contain the sequence 123, when a phone number has seven digits and cannot begin with a 0? Is there any other way to calculate this without using the manual way?

Answer 1

There are 24208 phone numbers that contain the sequence 123.

Explanation:

Yes, there are several ways to calculate the number of phone numbers containing the sequence 123 without using a manual approach.

Method: Inclusion-Exclusion Principle:

Define three events:

A1: The first three digits are 123.

A2: The second three digits are 123.

A3: The third three digits are 123.

Calculate the number of phone numbers satisfying each event individually:

N(A1) = 8 (first digit can be any number except 0) * 10 * 10 (last four digits can be any number) = 8000.

N(A2) = 10 * 8 (first two digits can be any number) * 10 (last five digits can be any number) = 8000.

N(A3) = 10 * 10 * 8 (first six digits can be any number) = 8000.

Apply the inclusion-exclusion principle:

N(123) = N(A1) + N(A2) + N(A3) - N(A1 ∩ A2) - N(A1 ∩ A3) - N(A2 ∩ A3) + N(A1 ∩ A2 ∩ A3).

Calculate the overlapping cases:

N(A1 ∩ A2) = 8 * 10 (last five digits can be any number) = 800.

N(A1 ∩ A3) = 10 * 8 (last four digits can be any number) = 800.

N(A2 ∩ A3) = 10 * 8 (last three digits can be any number) = 800.

N(A1 ∩ A2 ∩ A3) = 8 (first digit can be any number except 0) = 8.

Substitute all the values back into the inclusion-exclusion formula:

N(123) = 8000 + 8000 + 8000 - 800 - 800 - 800 + 8

= 24208

Thus, there are 24208 phone numbers that contain the sequence 123.