Question

A dance instructor chose four of his 10 students to be on stage for a performance.

If order does not matter, in how many different ways can the instructor choose the four students?

10 C 4 = StartFraction 10 factorial Over (10 minus 4) factorial 4 factorial

210
1,260
6,300
25,200

Answer 1

To determine how many different ways the instructor can choose 4 students from 10, we use the combination formula, which is given by:

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

where \( n \) is the total number of items to choose from (in this case, 10 students), and \( r \) is the number of items to choose (in this case, 4 students).

So, we can calculate \( \binom{10}{4} \) as follows:

\[
\binom{10}{4} = \frac{10!}{4!(10 - 4)!} = \frac{10!}{4!6!}
\]

Next, we can calculate \( 10! \), \( 4! \), and \( 6! \):

- \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)
- \( 6! = 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 720 \)

Now we only need the first 4 factors of \( 10! \) divided by \( 6! \):

\[
10! = 10 \times 9 \times 8 \times 7 \times 6!
\]

So:

\[
\binom{10}{4} = \frac{10 \times 9 \times 8 \times 7 \times 6!}{4! \times 6!} = \frac{10 \times 9 \times 8 \times 7}{4!}
\]
\[
= \frac{10 \times 9 \times 8 \times 7}{24}
\]

Calculating the numerator:

\[
10 \times 9 = 90
\]
\[
90 \times 8 = 720
\]
\[
720 \times 7 = 5040
\]

Now divide by \( 24 \):

\[
\frac{5040}{24} = 210
\]

Thus, the number of different ways the instructor can choose 4 students from 10 is **210**.

So the answer is:

\[
\boxed{210}
\]