The thirty-member Science Club is choosing a seven-member committee to determine which competitions to attend next year. How many different possible committees can be chosen? $\newline$ $a.$ $872,640$ $\newline$ $b.$ $593,775$ $\newline$ $c.$ $2,035,800$ $\newline$ $d.$ $658,008$ $\newline$ $e.$ $1,560,780$

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Q. The thirty-member Science Club is choosing a seven-member committee to determine which competitions to attend next year. How many different possible committees can be chosen?

\newline

a.

872,640

\newline

b.

593,775

\newline

c.

2,035,800

\newline

d.

658,008

\newline

e.

1,560,780

Identify Problem Type: Identify the type of problem. $\newline$ We need to find the number of ways to choose a seven-member committee from a thirty-member club. This is a combination problem because the order in which we select the committee members does not matter.
Use Combination Formula: Use the combination formula. $\newline$ The number of ways to choose $k$ members from a group of $n$ members is given by the combination formula $nCk = \frac{n!}{k! \cdot (n-k)!}$ , where $!$ denotes factorial.
Apply Formula to Problem: Apply the combination formula to the given problem. $\newline$ Here, $n = 30$ (total members) and $k = 7$ (members to choose). $\newline$ Calculate $30C7 = \frac{30!}{(7! * (30-7)!)}$ .
Simplify Factorials: Simplify the factorials. $\newline$ Calculate $\frac{30!}{7! \times 23!}$ by canceling out the common terms in the numerator and the denominator. $\newline$ $\frac{30!}{7! \times 23!} = \frac{30 \times 29 \times 28 \times 27 \times 26 \times 25 \times 24}{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$ .
Perform Calculations: Perform the calculations. $\newline$ Divide each term in the numerator by a corresponding term in the denominator to simplify the calculation: $\newline$ $\frac{30}{6} = 5$ $\newline$ $29$ is a prime number and cannot be simplified. $\newline$ $\frac{28}{7} = 4$ $\newline$ $\frac{27}{3} = 9$ $\newline$ $26$ is a prime number and cannot be simplified. $\newline$ $\frac{25}{5} = 5$ $\newline$ $\frac{24}{4} = 6$ $\newline$ Now multiply the simplified numbers together: $\newline$ $5 \times 29 \times 4 \times 9 \times 26 \times 5 \times 6 = 593,775.$
Check Answer Choices: Check the answer choices. $\newline$ The calculated number $593,775$ matches one of the given answer choices, which is option $b$ .