Question

What is the formula for calculating the number of combinations of selecting 3 students out of 15?

Answer 1

The formula for calculating combinations is given by:

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

where:
- \( n \) is the total number of items (in your case, students),
- \( r \) is the number of items to choose,
- \( ! \) denotes factorial, which is the product of all positive integers up to that number.

For your specific case of selecting 3 students out of 15, \( n = 15 \) and \( r = 3 \). Plugging these values into the formula, we have:

\[
C(15, 3) = \frac{15!}{3!(15-3)!} = \frac{15!}{3! \cdot 12!}
\]

Next, we can simplify the expression. The factorial \( 15! \) can be expressed as:

\[
15! = 15 \times 14 \times 13 \times 12!
\]

This allows us to cancel out \( 12! \) in the numerator and denominator:

\[
C(15, 3) = \frac{15 \times 14 \times 13}{3!}
\]

Now, calculate \( 3! \):

\[
3! = 3 \times 2 \times 1 = 6
\]

So, we have:

\[
C(15, 3) = \frac{15 \times 14 \times 13}{6}
\]

Calculating the numerator:

\[
15 \times 14 = 210
\]
\[
210 \times 13 = 2730
\]

Now divide by \( 6 \):

\[
C(15, 3) = \frac{2730}{6} = 455
\]

Therefore, the number of ways to select 3 students from 15 is:

\[
\boxed{455}
\]