There are $10$ people on a basketball team, and the coach needs to choose $5$ to put into a game. How many different possible ways can the coach choose a team of $5$ if each person has an equal chance of being selected? $\newline$ Answer:

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Grade 8

Probability of simple events

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Q. There are

10

people on a basketball team, and the coach needs to choose

5

to put into a game. How many different possible ways can the coach choose a team of

5

if each person has an equal chance of being selected?

\newline

Answer:

Calculate Combinations Formula: To solve this problem, we need to calculate the number of combinations of $10$ people taken $5$ at a time. The formula for combinations is given by: $\newline$ $C(n, k) = \frac{n!}{k! \cdot (n - k)!}$ $\newline$ where $n$ is the total number of items, $k$ is the number of items to choose, and $!$ denotes factorial. $\newline$ In this case, $n = 10$ and $k = 5$ .
Calculate Factorial of n: First, we calculate the factorial of $n$ , which is $10!$ . $10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$
Calculate Factorial of $k$ : Next, we calculate the factorial of $k$ , which is $5!$ . $5! = 5 \times 4 \times 3 \times 2 \times 1$
Calculate Factorial of $n - k$ : We also need to calculate the factorial of $n - k$ , which is $10 - 5 = 5$ , so we need $5!$ again. $\newline$ Since we've already calculated $5!$ , we can use the same value.
Plug Values into Formula: Now we can plug these values into the combinations formula: $\newline$ $C(10, 5) = \frac{10!}{5! \times (10 - 5)!}$ $\newline$ $C(10, 5) = \frac{10!}{5! \times 5!}$
Substitute Factorial Values: Substitute the factorial values we've calculated into the formula: $\newline$ $C(10, 5) = \frac{10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(5 \times 4 \times 3 \times 2 \times 1) \times (5 \times 4 \times 3 \times 2 \times 1)}$
Simplify Expression: We can simplify the expression by canceling out the common factors in the numerator and the denominator: $\newline$ $C(10, 5) = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1}$
Further Simplification: Further simplification gives us: $\newline$ $C(10, 5) = \frac{10}{5} \times \frac{9}{1} \times \frac{8}{4} \times \frac{7}{1} \times \frac{6}{2}$ $\newline$ $C(10, 5) = 2 \times 9 \times 2 \times 7 \times 3$
Multiply to Get Result: Multiplying these together, we get: $\newline$ $C(10, 5) = 2 \times 9 \times 2 \times 7 \times 3$ $\newline$ $C(10, 5) = 18 \times 14 \times 3$ $\newline$ $C(10, 5) = 252 \times 3$ $\newline$ $C(10, 5) = 756$