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 than can be made by adding two letters?

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 than can be made by adding two letters?

mathstudent55 · Answer

Original scheme: 5 * 3 * 4 = 60

Adding two letters to one set:
1) 7 * 3 * 4 = 84
2) 5 * 5 * 4 = 100
3) 5 * 3 * 6 = 90

Adding one letter to one set and one letter to another set:
1) 5 * 4 * 5 = 100
2) 6 * 3 * 5 = 90
3) 6 * 4 * 4 = 96

With the two new letters added, there can be a maximum of 84, 90, 96, and 100 different license plates. Therefore, the most additional license plates adding two letters allows is 100 - 60 = 40.