Final answer:
There are 2035800 different ways to choose an 8-member HOA board from a group of 21 homeowners (10 Eagle homeowners and 11 Quail homeowners). The calculation is performed using combinations where the order of selection does not matter.
Step-by-step explanation:
To find out the number of different ways there are to choose an 8-member HOA board from 10 Eagle homeowners and 11 Quail homeowners, we must use combinations since the order of selection does not matter. The total number of homeowners is 21 (10 Eagle homeowners + 11 Quail homeowners).
Since there are no restrictions given on the composition of the board in regards to Eagle or Quail homeowners, any of the 21 homeowners can be chosen for the 8 positions. This is a simple combinatorial problem, and the formula to calculate the number of combinations is C(n, k) = n! / (k!(n-k)!), where n is the total number of items to choose from, k is the number of items to choose, and ! denotes factorial.
So for our case, C(21, 8) = 21! / (8!(21-8)!) computes the total number of ways to choose the 8-member board.
Using this formula, we find:
- C(21, 8) = 21! / (8! * 13!)
- C(21, 8) = (21 * 20 * 19 * 18 * 17 * 16 * 15 * 14) / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
- C(21, 8) = 2035800 different ways.
Therefore, there are 2035800 different ways to choose an 8-member HOA board from the group of 21 homeowners.