Question

Number plates of cars must contains 3 letters of the alphabet denoting the place and area to which its owner belongs. This is to be followed by a three-digit number. How may different number plates can be formed if:
(i) repetition of letters and digits is not allowed.
(ii) rendition of letters and digits is allowed.

Answer 1

There are 26 letters of alphabet and 10 digits from 0 to 9,
Repetition not allowed :
3 letters $26\times 25\times 24$
3 digit numbers = ${}^{10}{P}_3{-}^9{P}_2$ (beginning with zero)
$=\mathrm{10.9.8}-9.8=9.8(10-1)=\mathrm{9.9.8}$
Number of plates = $26\times 25\times 24\times 9\times 9\times 8=10108800$. Repetition allowed :
3 letters $26\times 26\times 26$
3 digit numbers $=9\times 10\times 10=900$
1st place can't be filled by 0, 2nd can be filled by 10 and so is 3rd as repetition is allowed.
Number of plates = $26\times 26\times 26\times 900=15818400.$