Suppose that license plates in a certain country are made up of
three letters followed by 2
digits followed by 3 more letters. How many license plates can be made if no repetitions are allowed within each set. (That is, no repetition within the first set of letters, the set of digits, or the last set of letters. But, for instance, there could be an "A" in the first set of letters and an "A" within the last set of letters)
3 letters ... no rep ... 26 * 25 * 24
2 digits ... no rep ... (00 to 99) - 10
number of plates
... [(26 * 25 * 24)^2] * (100 - 10)
To find the number of license plates that can be made, we need to consider the possible choices for each set of characters.
For the first set of letters, since no repetitions are allowed, we have to choose 3 letters out of 26 (since there are 26 letters in the alphabet). We can use the formula for choosing without repetition, which is nCr = n! / (r! * (n-r)!), where n is the total number of elements and r is the number of elements to choose. In this case, n = 26 and r = 3. So the number of choices for the first set of letters is 26! / (3! * (26-3)!) = 26! / (3! * 23!).
For the set of digits, we have to choose 2 digits out of 10 (since there are 10 digits from 0 to 9). Using the same formula as before, the number of choices for the set of digits is 10! / (2! * (10-2)!).
Finally, for the last set of letters, we have to choose 3 letters out of 26, just like the first set. So, the number of choices for the last set of letters is also 26! / (3! * 23!).
To find the total number of license plates that can be made, we need to multiply the number of choices for each set together. So, the total number of license plates is:
(26! / (3! * 23!)) * (10! / (2! * 8!)) * (26! / (3! * 23!))
Simplifying this expression will give us the final answer.