Question

The serial number on a bill consists of a letter followed by nine digits and then a letter. How many different serial numbers

are possible, given the following conditions?
(a) Letters and digits cannot be repeated.
(b) Letters and digits can be repeated.
(c) The letters are nonrepeated consonants and the digits can be repeated. (Let y be a consonant.)

Answer 1

(a) 2,358,720,000

(b) 676,000,000,000

(c) 420,000,000,000

Explanation:

The serial number on a bill consists of a letter followed by nine digits and then a letter.

There are 26 different letters and 10 digits (0 through 9) available.

Part (a)

If the letters and digits cannot be repeated, there are:

• 26 choices for the first letter and 25 choices for the second letter.

• 10 choices for the first digit, 9 choices for the second digit, 8 choices for the third digit, and so on.

Therefore, the number of serial numbers that are possible if the letter and digits cannot be repeated is:

26 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 25 = 2,358,720,000

Part (b)

If the letters and digits can be repeated, there are:

• 26 choices for each letter.

• 10 choices for each digit.

Therefore, the number of serial numbers that are possible if the letter and digits can be repeated is:

26 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 26 = 676,000,000,000

Part (c)

There are 21 consonants in the alphabet.

If the letters are non-repeated consonants and the digits can be repeated, there are:

• 21 choices for the first letter and 20 choices for the second letter.

• 10 choices for each digit.

Therefore, the number of serial numbers that are possible if the letters are non-repeated consonants and the digits can be repeated is:

21 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 10 × 20 = 420,000,000,000