Question

How many 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 not allowed?

Answer 1

(i) Zero cannot be first digit of the license plates.
This mean the first digit can be selected from the 9 digits 1,2,3,4 ..., 9
So, there are 9 ways of filling the first digit of the license plates.
Now, 9 digits are left including 0. So, second place can be filled with any of the remaining 9 digits in 9 ways.
The third place of the license plates can be filled with in any of the remaining 7 digits.
So, there are 7 ways of filling the fourth place.
Hence, the total number of ways
= $9\times 9\times 8\times 7\times 6=27216$
(ii) Zero cannot be first digit of the license plates.
$\therefore$ first digit can be selected from the 9 digits 1,2,3,...., 9
So, there are 9 ways of filling the first digit of the licnse plates.
The repetition of digits is allowed to made a license plates number.
$\therefore$ The number of ways to fill the remaining places of the number.
= $10\times 10\times 10\times 10$.
Hence, the total nubmer of ways
= $9\times 10\times 10\times 10\times 10$ = 90,000