Question

movie selections the foreign language club is showing a five-movie marathon of subtitled movies. how many ways can they choose 5 from the 17 available? there are □ ways to choose 5 from the available 17 movies.

Explanation:

Step1: Identify the problem type

This is a combination problem since the order of choosing the movies does not matter. The formula for combinations is \( C(n, k)=\frac{n!}{k!(n - k)!} \), where \( n = 17 \) (total number of movies) and \( k = 5 \) (number of movies to choose).

Step2: Calculate factorial values

First, calculate \( n!=17! = 17\times16\times15\times14\times13\times12\times\cdots\times1 \), \( k!=5! = 5\times4\times3\times2\times1 = 120 \), and \( (n - k)!=(17 - 5)!=12! = 12\times11\times\cdots\times1 \).

Step3: Simplify the combination formula

Using the combination formula \( C(17, 5)=\frac{17!}{5!(17 - 5)!}=\frac{17!}{5!×12!} \). We can simplify this as \( \frac{17\times16\times15\times14\times13\times12!}{5!×12!} \). The \( 12! \) terms cancel out, leaving \( \frac{17\times16\times15\times14\times13}{5\times4\times3\times2\times1} \).

Step4: Perform the arithmetic

Calculate the numerator: \( 17\times16 = 272 \), \( 272\times15 = 4080 \), \( 4080\times14 = 57120 \), \( 57120\times13 = 742560 \).
Calculate the denominator: \( 5\times4 = 20 \), \( 20\times3 = 60 \), \( 60\times2 = 120 \), \( 120\times1 = 120 \).
Now divide the numerator by the denominator: \( \frac{742560}{120}=6188 \).

Answer:

\( 6188 \)