So I've been faced with a question by one of my friends and I'm not a math major, however I'm trying to figure it out.
If we know that license plates are "6 characters long" and consist of either numbers (0-9) and letters (a-z) lowercase. How can I determine the number of ways to determine the following.
*letters and numerals can be repeated?
My Guess: 36 Permute 6
*each letter and numeral can be used at most once?
My Guess: 36 Choose 6
*the license plate must have a letter as its first character and each letter or numeral can be used at most once?
My Guess:[(10 choose 5)(26 choose 1)(25 choose 5)]/(36 choose 6)
Best Answer
If letters and numerals can be repeated, it will in fact be $36^6$. For each of the six slots, we have 36 choices for what can go there. By the rule of product, there are then $36\cdot 36\cdot 36\cdot 36\cdot 36\cdot 36=36^6$ different such license plates.
If letters and numerals cannot be repeated, you will have 36 choices for the first slot, 35 choices for the second, 34 for the third,... for a total of $36\cdot 35\cdot 34\cdot 33\cdot 32\cdot 31 = \frac{36!}{30!}$
I'll leave the final one to you to think about, but again, try to break it up via rule of product by figuring out how many choices there are for the first spot, how many choices for the second, how many choices for the third, etc... and multiply those numbers together.