How many possible license plates can be manufactured if a license plate consists of three letters followed by three digits and (a) the digits must be distinct; the letters can be arbitrary? (b) the letters must be distinct; the digits can be arbitrary? (c) the digits and the letters must be distinct?

(a) 12,654,720 different plates

(b) 15,600,000 different plates

(c) 11,232,000 different plates

Step-by-step explanation:

There are 3 letters followed by 3 digits. For each letter, there are 26 possibilities, and for each digit there are 10 possibilities.

(a)

If the digits must be distinct, the first digit has 10 possibilities, the second one has only 9 (1 was already taken) and the third digit has only 8, so the number of different license plates is:

26*26*26*10*9*8 = 12,654,720 different plates

(b)

If the letters must be distinct, the first letter has 26 possibilities, the second one has only 25 (1 was already taken) and the third letter has only 24, so the number of different license plates is:

26*25*24*10*10*10 = 15,600,000 different plates

(c)

If both the letters and digits must be distinct, we have that number of different license plates is:

26*25*24*10*9*8 = 11,232,000 different plates