License plates in a particular state display 22 letters followed by 44 numbers. how many different license plates can be manufactured for this state?

Since there are 26 letter choices for the first spot, and 26 for the next, etc. we get 26^22 for the first 22 letters. For the 44 numbers, we have 10^44 due to that there are 10 options for the first number, and it doesn't matter what it is - there's still 10 options for the next number. In addition to that, for each number you put in the first slot, there are 10 ways to put numbers in the second slot, etc..

26^22*10^44 is your answer  due to that if you think of it as 2 numbers in a row at this point, there are 10^44 different numbers for each different letter change in those 26^22.