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📅 March 1, 2026 👤 scraface

Mar 01, 2026

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📌 A primatologist is observing a group of 10 primates and wants to form pairs for a study. How many distinct pairs of primates can be formed?

📌 To determine the number of distinct pairs, we use combinations to choose 2 primates from 10. This is calculated as \(\binom{10}{2}\).

📌 Substituting \(n = 10\) and \(r = 2\):

📌 \binom{10}{2} = \frac{10!}{2!(10-2)!} = \frac{10 \times 9}{2 \times 1} = 45

📌 Thus, the number of distinct pairs of primates is \(\boxed{45}\).

📌 An entomologist is cataloging insects and wants to select 3 from a group of 12 different species for a detailed genetic analysis. How many ways can the entomologist choose the species?

📌 The number of ways to choose 3 species from 12 is calculated using the combination formula \(\binom{12}{3}\).

📌 \binom{12}{3} = \

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Use the Law of Cosines: \( c^2 = a^2 + b^2 - 2ab\cos C \), where \( a...
\( c^2 = 13^2 + 15^2 - 2 \cdot 13 \cdot 15 \cdot 0.5 = 169 + 225 - 195...
So, \( c = \sqrt{199} \approx 14.106 \), but we keep exact: \( \sqrt{1...
However, for exact boxed answer: \( \sqrt{199} \) is simplest.
But compute: \( 169 + 225 = 394 \), \( 2 \cdot 13 \cdot 15 = 390 \), \...
Thus, \( c = \sqrt{199} \).
#### \(\sqrt{199}\)
A circle is inscribed in an equilateral triangle with side length 12 c...
In an equilateral triangle, the inradius \( r = \frac{s\sqrt{3}}{6} \)...
So, \( r = \frac{12\sqrt{3}}{6} = 2\sqrt{3} \).

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