Question

Mixed species flocks of birds include social and solitary species. There are 5 social species and 10 solitary species in a forest. Flocks always have a total of 5 species, of which 2 are social and 3 are solitary. The number of types of flocks with unique species composition is ___________. (Answer in integer)

Answer 1

Correct Answer: 1200

Step 1: Interpret “types of flocks with unique species composition.”
Order does \emph{not} matter (a set of species). We choose subsets: \(2\) from the \(5\) social species, and \(3\) from the \(10\) solitary species — independent choices.

Step 2: Count choices with combinations.
\[ \binom{5}{2} = \frac{5!}{2!\,3!}=10, \qquad \binom{10}{3} = \frac{10!}{3!\,7!}=120. \]

Step 3: Multiply independent categories.
\[ \text{Total distinct compositions} = \binom{5}{2}\times \binom{10}{3} = 10\times 120 = \boxed{1200}. \]

Step 4: Reasonableness check.
Maximum possible 5-species sets from \(15\) species is \(\binom{15}{5}=3003\). Our constraint (exactly \(2\) social \(+\) \(3\) solitary) reasonably narrows this to \(1200\).