Question

Bicycle license plates in Flatville each contain three letters. The first is chosen from the set $\{C,H,L,P,R\},$ the second from $\{A,I,O\},$ and the third from $\{D,M,N,T\}.$

When Flatville needed more license plates, they added two new letters. The new letters may both be added to one set or one letter may be added to one set and one to another set. What is the largest possible number of additional license plates that can be made by adding two letters?

$24$

$30$

$36$

$40$

$60$

Solution:

There are currently $5 \cdot 3 \cdot 4 = 60$ license plates that can be made.

If both letters are added to the first set, then there are $7 \cdot 3 \cdot 4 = 84$ possible plates.

If they are both added to the second, there are $5 \cdot 5 \cdot 4 = 100$ plates.

If they are added to the third, there are $5 \cdot 3 \cdot 6 = 90$ choices.

If one is added to the first set and the other to the second set, there are $6 \cdot 4 \cdot 4 = 96$ plates.

If the other is added to the third set, we get $6 \cdot 3 \cdot 5 = 90$ possible plates.

Finally, if the letters are added to the second and third sets, there are $5 \cdot 4 \cdot 5 = 100$ plates.

We see that $100$ is the greatest number of plates that we can achieve. This is an additional $100 - 60 = 40$ plates.

Thus, D is the correct answer.

1999 AMC 8 Problem 15

Solution:

Problem 15 in Other Years