Answer:
9,100,000
Explanation:
First, let's figure out how many license plates begin with 2 and end with A.
There are 7 positions total in the license plate (5 + 2). For the first position, there is only one option, since the first digit must be 2. For the following 4 positions, there are 10 options for each position, since there are 10 one-digit numbers (0 to 9).
For the next position (the 6th position), there are 26 options, since it can be any letter in the alphabet. For the last position, there is only option, since the last letter must be A.
We multiply the number of options for each position by the options for all the other positions (because of the multiplication principal), which gives us:
1 x 10 x 10 x 10 x 10 x 26 x 1 = 260,000
Second, let's figure out how many license plates start with 2 (but don't necessarily end in A):
1 x 10 x 10 x 10 x 10 x 26 x 26 = 6,760,000
Third, let's figure out how many license plates end in A (but don't necessarily start with 2):
10 x 10 x 10 x 10 x 10 x 26 x 1 = 2,600,000
Now, to figure out how many license plates begin with 2 or end with A, we can first add:
how many license plates start with 2 + how many license plates end in A:
= 6,760,000 + 2,600,000 = 9,360,000
But, why did we bother doing our first calculation, of how many license plates begin with 2 and end with A? Well, the answer 9,360,000 is not correct, because it overcounted the possibilities. How? Well...
- the number of license plates that start with 2 included all the license plates that start with 2 and end in A
- the number of license plates that end in A included all the license plates that end in A and start with 2
See the problem? When we added these two numbers together, we added all the ways in which a license plate can start with 2 and end in A twice. Therefore, we need to subtract it once, so that we do not overcount.
So, we can now do:
(how many license plates start with 2 + how many license plates end in A)
- how many license plates start with 2 and end in A
= 6,760,000 + 2,600,000 - 260,000 = 9,100,000