Question

Use combinations to solve the following problem. From a class of 17 students, the teacher needs 5 students to work in the student store during lunch. How many ways can the teacher make the schedule with 5 different students?(1 point)
Responses

75
75

1
1

6,188
6,188

742,560

Answer 1

To find the number of ways to choose 5 students from a class of 17 students, we can use the combination formula, which is represented as:

\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]

where \( n \) is the total number of students (17 in this case), \( r \) is the number of students to choose (5 in this case), and \( ! \) denotes factorial.

Using this formula:

\[
\binom{17}{5} = \frac{17!}{5!(17-5)!} = \frac{17!}{5! \cdot 12!}
\]

Calculating this step-by-step:

1. Calculate \( 17! \) up to \( 12! \):

\[
\frac{17 \times 16 \times 15 \times 14 \times 13 \times 12!}{5! \times 12!}
\]

2. The \( 12! \) terms cancel out:

\[
\frac{17 \times 16 \times 15 \times 14 \times 13}{5!}
\]

3. Now calculate \( 5! = 5 \times 4 \times 3 \times 2 \times 1 = 120 \).

4. Now compute the product of the numerator:

\[
17 \times 16 = 272,
\]
\[
272 \times 15 = 4080,
\]
\[
4080 \times 14 = 57120,
\]
\[
57120 \times 13 = 742560.
\]

5. Finally, divide by \( 120 \):

\[
\frac{742560}{120} = 6188.
\]

So the number of ways to choose 5 students from a class of 17 is:

\[
\binom{17}{5} = 6188.
\]

Therefore, the correct answer is **6,188**.