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

( 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

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

8585

To get the answers to these questions, we need to understand the concept of permutations and combinations.

(i) To find the number of possible distinct license plates, we need to calculate the total number of arrangements for the given format.

The first three positions can be filled with any letter, so we have 26 choices for each position. (26 letters in the English alphabet) Thus, the number of choices for the three letters is 26 * 26 * 26 = 26^3.

Similarly, the last four positions can be filled with any digit, so we have 10 choices for each position. (10 digits from 0 to 9) Therefore, the number of choices for the four digits is 10 * 10 * 10 * 10 = 10^4.

To find the total number of distinct license plates, we multiply the number of choices for the letters and the number of choices for the digits: 26^3 * 10^4.

(ii) If the letters O and I and the digits 0 and 1 are prohibited, we subtract these choices from the total number of choices we calculated in part (i).

Since there are 24 other letters and 6 other digits to choose from, the number of choices for the letters becomes 24 * 24 * 24, and the number of choices for the digits becomes 6 * 6 * 6 * 6.

The total number of distinct license plates with prohibited characters is 24^3 * 6^4.

(iii) To find the number of different license plates with precisely two C's and two 2's, we need to consider placing the two C's in position and then filling in the other letter. A similar approach can be used for the digits.

There are 3 positions available for the two C's (assuming the letters are distinct), and we can choose 2 out of these 3 positions in C(3, 2) ways. (C(n, r) denotes the combination formula n choose r)

For the remaining letter, we have 26 choices.

Similarly, there are 4 positions available for the two 2's, and we can choose 2 out of these 4 positions in C(4, 2) ways. For the remaining digits, we have 10 choices.

To find the total number of different license plates, we multiply the number of choices for the two C's, the number of choices for the remaining letter, the number of choices for the two 2's, and the number of choices for the remaining digits: C(3, 2) * 26 * C(4, 2) * 10.

Simplifying the expression gives us the final answer.

Note: Make sure to apply the correct combination formulas (n choose r) and compute the values accurately to get the correct results.