(a) how many different 7-place license plates are possible if the first 2 places are for letters and the other 5 for numbers?
(b) repeat part (a) under the assumption that no letter or number can be repeated in a single license plate.

a) 67600000 possible plates

b) 19656000 possible plates where neither digit or letter are repeated

Step-by-step explanation:

The easiest way to solve this problem is seeing how many possibilities and there for each place in the license plate:

We have the following 7 places:

P1 - P2 - P3 - P4 - P5 - P6 - P7

The alphabet has 26 letters and there are 10 digits. So,

a) For each position there are the following possibilites

26 - 26 - 10 - 10 - 10 - 10 - 10

So in all, there are, 26*26*10*10*10*10*10 = 67600000 possible plates

b) Let's do the possibilites for each position again. Now, no letter or number can be repeated, so.

P1 is still 26. Now for P2, the letter in position P1 cannot be repeated, so there are only 25 possibilies. As for the digits, for P3, the first digit, there are still 10 possibilities. For P4, there are 9, since the digit in P3 cannot be repeated. For P5 there are 8, since P3 and P4 cannot be repeated... So there are the following number of possibilities:

26-25-10-9-8-7-6

In all, there are 26*25*10*9*8*7*6 = 19656000 possible plates where neither digit or letter are repeated.

nlgqer

step-by-step

c. 1,233 miles