I have looked all over the web for some additional information on this matter with no results.
Let's say a new form of license plate have 4 letters followed by 3 digits and all sequences are possible.
My understanding is since there are 52 letters (lower case and upper case) and 10 digits then the possibilities are $52^4 \cdot 10^3 = 7311616000$ possibilities for the entire license plate with no specific rules.
The issue I am running into now is when certain rules are applied:
1) The license plate must have exactly two T's and end in one 5. The solution stated in the book is 375,000.
I have tried $52^2 \cdot 1 \cdot 1 \cdot 10^2 \cdot 1$, but it is not equal to 375,000.
2) This is another question but requires at least one "T" and the digit "4" someplace in them. The solution in the book states it is 17,981,121.
The second question came to a halt as well as I used the same method in question 1.
Is there something I am doing incorrectly because the answer is not meeting with the solution.
Best Answer
The first problem is that the author is considering lower case and upper case to be the same. So you should switch $52$ for $26$ in all your arguments
For the first question you are assuming that the two $T$'s are the last two letters, while this is not necessarily the case, to fix this first select the two positions that will have the $T$ (There are $6$ ways to do this). And after that use your argument, only you need to modify it since there are in fact $25$ options for each of the non-$T$ letters (Since $T$ is not allowed). So the answer should be $6\cdot25^2\cdot10^2\cdot1=375000$ as desired.
For the second question there are in fact $26^4\cdot10^3=456976000$ total Plates without any restrictions (I just used your argument switching $52$ for $26$).
Instead of counting the Plates that have at least one $T$ and at least on $4$ we shall count those that are lacking a $4$ or a $T$, and then substract this number from the total.
How many don't have a $4$ or don't have a $T$? We count those that don't have a $4$ and we count those that don't have a $T$ and add these numbers, and we subtract those that have no $4$ and no $T$ because those sequences were counted two times.(This idea is sometimes called inclusion-exclusion).
How many have no $T$? If they don't have $T$ there are only $25$ options for each letter, so there are $25^4\cdot10^3=390625000$
How many have no $4$? If they don't have $4$ there are only $9$ options for each digit, so there are $26^4\cdot 9^3=333135504$
How many don't have $T$ and don't have a $4$? There are $25$ options for each letter and $9$ options for each digit so $25^4\cdot9^3=284765625$.
So by our previous argument there are $390625000+333135504-284765626=438994879$ Plates that have no $4$ or no $T$. And so we have $456976000-438994879=17981121$ that have at least one $T$ and at least one $4$.