A small auditorium labels its seats with one of the 26 a-z letters and one of the ten digits 0-9. How many di↵erent seat labels are possible? What is the answer if each seat is labeled with one of the letters a-z and one of the number 1-100? What is the answer if each seat is labeled with two letters and two digits?

a. 520 seats

b. 5200 seats

c. 67600 seats

Explanation:

a. How many different seat labels are possible?

Given

Number of Alphabet = 26

Number of digits = 10

The auditorium label its seats with one alphabet and one digits.

This mean that the auditorium uses two characters to label its seats.

The auditorium may decide to start with an alphabet followed by a digit or a digit then followed by an alphabet

this is given by:

(Alphabet and Digit) or (Digit and Alphabet)

Number of seats = (26 * 10) + (10 * 26)

= 260 + 260

= 520 seats

b. What is the answer if each seat is labeled with one of the letters a-z and one of the number 1-100?

Given

Number of Alphabet = 26

Number of digits = 100

The number are treated individually (I.e 10 is one number, not two numbers as in 1 and 0).

The auditorium may decide to start with an alphabet followed by a number or a number then followed by an alphabet

this is given by:

(Alphabet and Number) or (Number and Alphabet)

Number of seats = (26 * 100) + (100 * 26)

= 2600 + 2600

= 5200 seats

c. What is the answer if each seat is labeled with two letters and two digits?

Here, well assume that repetition of digits and alphabets are allowed

Given

Number of alphabets = 26

Number of digits = 10

Number of seats is given by:

2 alphabets and 2 digits

= 26 * 26 * 10 * 10

= 67600 seats

because we are smart