Criterion Channel Stream: Exclusive Premiere You Need to Watch Tonight!
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5Question: A primatologist observes 7 different chimpanzees engaging in a communication ritual where each chimpanzee either vocalizes or gestures, but not both, and at least two chimpanzees must gesture. How many distinct communication patterns are possible?
Solution: Each chimpanzee has 2 choices: vocalize or gesture, but since each must choose exactly one and at least two must gesture, we begin by computing the total number of unrestricted assignments and subtract invalid cases.
Total assignments (each chimp independently chooses vocalize or gesture):
2^7 = 128
Invalid cases:
0 gesturing (all vocalize): $ \binom{7}{0} = 1 $
1 gesturing: $ \binom{7}{1} = 7 $
So, number of invalid patterns: $ 1 + 7 = 8 $
Valid communication patterns: