# How many different five-digit number licence plates can be made if

Question:

How many different five-digit number licence plates can be made if

(i) first digit cannot be zero and the repetition of digits is not allowed,

(ii) the first-digit cannot be zero, but the repetition of digits is allowed?

Solution:

(i) Since the first digit cannot be zero, the number of ways of filling the first digit = 9

Number of ways of filling the second digit = 9     (Since repetition is not allowed)

Number of ways of filling the third digit = 8

Number of ways of filling the fourth digit = 7

Number of ways of filling the fifth digit = 6

Total number of licence plates that can be made $=9 \times 9 \times 8 \times 7 \times 6=27216$

(ii) Since the first digit cannot be zero, the number of ways of filling the first digit = 9

Number of ways of filling the second digit = 10    (Since repetition is allowed)

Number of ways of filling the third digit = 10

Number of ways of filling the fourth digit = 10

Number of ways of filling the fifth digit = 10

Total number of licence plates that can be made $=9 \times 10 \times 10 \times 10 \times 10=90000$