The current license plates in New York State consist of three letters followed by four digits.

Question

( i ) How many possible distinct license plates can there be?
( ii ) How many possible distinct license plates could there be if the letters O and I and the digits, 0 and 1 were prohibited because they look alike?
( iii ) How many different license plates are there that have precisely two C's and two 2's? (Here we are no longer prohibiting O's, I's, 0's and 1's.) [Hint: Consider placing the two C's in position and then filling in the other letter. Do something similar with the digits

Answer 1

3 letters ---> 26^3
3 numbers, assuming it may start with a zero
--- 10^4

number of plates = 26^3 (10^4) = 175760000

(about 176 million, long way to go)

ii) so only 24 letters, and 8 numbers
-- what do you think?

iii) look at the letter part first:
assume we put CC at the front
so number of such cases = 1x1x25x25 = 625
CCMP could be one those, but each one can be arranged in 4!/2! or 12 ways
the letters can be arranged in 625x12 or 7500 ways
using the same argument for the numbers, there are
1x1x9x9 x 12 ways or 972 ways
number of ways according to the stated conditions
= 7500x972 = 7290000 ways

Answer 2

8585