Calculating Unique Bird Flock Species Composition

The problem asks for the number of different ways to form a mixed species flock with a specific composition from a larger group of species.

Given Information:

• Number of social species available: 5
• Number of solitary species available: 10
• Required social species in a flock: 2
• Required solitary species in a flock: 3
• Total species in a flock: 5 (2 social + 3 solitary)

Combinations Approach

Since the order in which species are selected does not matter, this is a problem of combinations. We need to calculate the number of ways to choose the social species and the number of ways to choose the solitary species separately, and then multiply them.

1. Selecting Social Species

We need to choose 2 social species from the 5 available. The number of combinations is calculated using the formula $C(n, k) = \binom{n}{k} = \frac{n!}{k!(n-k)!}$.

Number of ways to choose social species = $C(5, 2) = \binom{5}{2}$

$ C(5, 2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!} = \frac{5 \times 4}{2 \times 1} = 10 $

2. Selecting Solitary Species

We need to choose 3 solitary species from the 10 available.

Number of ways to choose solitary species = $C(10, 3) = \binom{10}{3}$

$ C(10, 3) = \frac{10!}{3!(10-3)!} = \frac{10!}{3!7!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 10 \times 3 \times 4 = 120 $

3. Total Unique Flock Compositions

To find the total number of unique flock types, we multiply the number of ways to select the social species by the number of ways to select the solitary species (using the multiplication principle of counting).

Total unique flocks = (Ways to choose social species) $\times$ (Ways to choose solitary species)

$ \text{Total Flocks} = 10 \times 120 = 1200 $

Conclusion

There are 1200 different types of flocks with unique species composition possible.