\binom{n}{r} = \frac{n!}{r!(n-r)!}

February 22, 2026

$\binom{n}{r} = \frac{n!}{r!(n-r)!}$

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A primatologist is observing a group of 6 primates where each primate communicates with every other primate exactly once per day. How many unique communication events occur in one day?
To determine the number of unique communication events, we need to find the number of ways to choose 2 primates out of 6 to form a communication event. This is a combination problem where we choose 2 out of 6, denoted as \(\binom{6}{2}\).
The formula for combinations is given by:
Substituting \(n = 6\) and \(r = 2\):
\binom{6}{2} = \frac{6!}{2!(6-2)!} = \frac{6 \times 5}{2 \times 1} = 15
Thus, the number of unique communication events in one day is \(\boxed{15}\).

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