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While organizing a film festival, you must decide which of the 14 movies will be shown on the big screen. You only have the budget to show 7 movies on this screen, and you want to be able to tell moviegoers which order the films will be shown. In how many different ways can you show 7 of the 14 movies on the big screen?

While organizing a film festival, you must decide which of the $14$ movies will be shown on the big screen. You only have the budget to show $7$ movies on this screen, and you want to be able to tell moviegoers which order the films will be shown. In how many different ways can you show $7$ of the $14$ movies on the big screen?

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Compound events: find the number of outcomes

Full solution

Q. While organizing a film festival, you must decide which of the

14

movies will be shown on the big screen. You only have the budget to show

7

movies on this screen, and you want to be able to tell moviegoers which order the films will be shown. In how many different ways can you show

7

of the

14

movies on the big screen?

Identify the problem type: Identify the problem type. $\newline$ We need to find the number of ways to choose and arrange $7$ movies out of $14$ . This is a permutation problem because the order in which movies are shown matters.
Calculate the number of permutations: Calculate the number of permutations. $\newline$ The number of ways to arrange $7$ movies out of $14$ is given by the permutation formula $P(n, k) = \frac{n!}{(n-k)!}$ , where $n$ is the total number of items to choose from, and $k$ is the number of items to arrange. $\newline$ Here, $n = 14$ and $k = 7$ . $\newline$ $P(14, 7) = \frac{14!}{(14-7)!} = \frac{14!}{7!}$
Simplify the calculation: Simplify the calculation. $\newline$ $\frac{14!}{7!} = 14 \times 13 \times 12 \times 11 \times 10 \times 9 \times 8$ (since the terms $7!$ in the numerator and denominator cancel each other out). $\newline$ $= 2162160$

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