To determine the number of ways the dance instructor can choose 4 students out of 10, we use the concept of combinations. The formula for combinations (denoted as [tex]\( \binom{n}{k} \)[/tex] or [tex]\( nCk \)[/tex]) is as follows:
[tex]\[
\binom{n}{k} = \frac{n!}{k!(n-k)!}
\][/tex]
Here, [tex]\( n \)[/tex] represents the total number of students, which is 10, and [tex]\( k \)[/tex] represents the number of students to be chosen, which is 4. Plugging these values into the formula, we have:
[tex]\[
\binom{10}{4} = \frac{10!}{4!(10-4)!} = \frac{10!}{4! \cdot 6!}
\][/tex]
The factorial notation [tex]\( n! \)[/tex] (read as "n factorial") represents the product of all positive integers up to [tex]\( n \)[/tex]. Therefore, we calculate the factorial values as follows:
[tex]\[
10! = 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1
\][/tex]
[tex]\[
4! = 4 \times 3 \times 2 \times 1 = 24
\][/tex]
[tex]\[
6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720
\][/tex]
Substituting these factorial values into the combination formula:
[tex]\[
\binom{10}{4} = \frac{10!}{4! \cdot 6!} = \frac{3,628,800}{24 \cdot 720} = \frac{3,628,800}{17,280} = 210
\][/tex]
Thus, the number of ways the instructor can choose 4 students out of 10, without considering the order, is 210.
So, the correct answer is:
[tex]\[
210
\][/tex]
