Problem:
How many 15-letter arrangements of 5 A's, 5 B's, and 5 C's have no A's in the first 5 letters, no B's in the next 5 letters, and no C's in the last 5 letters?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Since the first group of five letters contains no A's, it must contain B's and C's for some integer with . Since the third group of five letters contains no C's, the remaining C's must be in the second group, along with A's.
Similarly, the third group of five letters must contain A's and ( ) B's. Thus each arrangement that satisfies the conditions is determined uniquely by the location of the B's in the first group, the C's in the second group, and the A's in the third group.
For each , the letters can be arranged in ways, so the total number of arrangements is
The problems on this page are the property of the MAA's American Mathematics Competitions