To determine how many different ways the dance instructor can choose four students out of ten, where the order does not matter, we use the concept of combinations.
The formula for combinations, also known as the binomial coefficient, is given by:
[tex]\[
{}_{n}C_{k} = \frac{n!}{k!(n-k)!}
\][/tex]
In this context:
- [tex]\( n \)[/tex] is the total number of students, which is 10.
- [tex]\( k \)[/tex] is the number of students to choose, which is 4.
Substitute the given values into the formula:
[tex]\[
{}_{10}C_{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4!6!}
\][/tex]
Where:
- [tex]\( 10! \)[/tex] (10 factorial) is the product of all positive integers up to 10.
- [tex]\( 4! \)[/tex] (4 factorial) is the product of all positive integers up to 4.
- [tex]\( 6! \)[/tex] (6 factorial) is the product of all positive integers up to 6.
We simplify this step-by-step:
1. Calculate [tex]\( 10! \)[/tex]:
[tex]\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
2. Calculate [tex]\( 4! \)[/tex]:
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
3. Calculate [tex]\( 6! \)[/tex]:
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
Next, substitute these values back into the combination formula:
[tex]\[
{}_{10}C_{4} = \frac{10!}{4!6!} = \frac{10!}{24 \times 720}
\][/tex]
We know [tex]\( 10! \)[/tex] is a large number, but it cancels out in part when divided by [tex]\( 6! \)[/tex]. Specifically, we can consider the first four terms of [tex]\( 10! \)[/tex]:
[tex]\[
10 \times 9 \times 8 \times 7 = 5040
\][/tex]
So we simplify [tex]\( \frac{5040}{24} \)[/tex]:
[tex]\[
{}_{10}C_{4} = \frac{5040}{24} = 210
\][/tex]
Therefore, the number of ways the instructor can choose four students out of ten is:
[tex]\[
210
\][/tex]
Hence, the answer is 210.
