Cherese started a Green Club at her school and needs to appoint a board of members. There are eight people available for these roles. How many different selections of four out of eight people could she choose? Write Pascal's Triangle from row 0 to row 8 and use it to solve the problem. (1 point)

1 answer

To determine how many different selections of four people can be chosen from eight, we can use the binomial coefficient, commonly denoted as \( \binom{n}{r} \), which represents the number of ways to choose \( r \) elements from a set of \( n \) elements without regard to the order of selection. In this case, we need to calculate \( \binom{8}{4} \).

First, let's construct Pascal's Triangle up to row 8:

Row 0:             1
Row 1:           1   1
Row 2:         1   2   1
Row 3:       1   3   3   1
Row 4:     1   4   6   4   1
Row 5:   1   5  10  10   5   1
Row 6: 1   6  15  20  15   6   1
Row 7: 1   7  21  35  35  21   7   1
Row 8: 1   8  28  56  70  28   8   1

In Pascal's Triangle, each entry \( \binom{n}{r} \) corresponds to the \( r \)-th entry of the \( n \)-th row, starting from row 0.

  • Row 8:
    • \( \binom{8}{0} = 1 \)
    • \( \binom{8}{1} = 8 \)
    • \( \binom{8}{2} = 28 \)
    • \( \binom{8}{3} = 56 \)
    • \( \binom{8}{4} = 70 \)
    • \( \binom{8}{5} = 56 \)
    • \( \binom{8}{6} = 28 \)
    • \( \binom{8}{7} = 8 \)
    • \( \binom{8}{8} = 1 \)

From the constructed Pascal's Triangle, we find that \( \binom{8}{4} = 70 \).

Thus, the total number of different selections of four out of eight people that Cherese can choose is \( \boxed{70} \).