How many 7place license plates are possible if the first thr

How many 7–place license plates are possible if the first three places are for letters and the last four places are are for numbers and: a) repeats are allowed in both letters and numbers; b) repeats are not allowed in the letters but are allowed in the numbers; c) repeats are not allowed in the letters or the numbers?

Solution

a)

There are 26 letters and 10 digits. Thus, as repetition is allowed, there are

#ways = (26^3)*(10^4) = 175760000 [answer]

b)

As repeats are not allowed for letters, then there are now just 26P3 ways to choose letters. Hence,

#ways = (26P3)*(10^4) = 156000000 [answer]

c)

As repeats are not allowed for letters and numbers, then there are now just 26P3 ways to choose letters, and 10P4 ways to choose numbers. Hence,

#ways = (26P3)*(10P4) = 78624000 [answer]