Question

Mrs. Plum’s civics class is going on a field trip to observe the state legislature while it's in session. There are 24 students in the class. The gallery where the class will sit has seating for three people in the front row. In how many ways can Mrs. Plum combine three of the students to sit in the front row if the order is not important?

12,144

1,012

6,072

2,024

Answer 1

To determine the number of ways Mrs. Plum can combine three students from the class of 24 to sit in the front row, we need to calculate the number of combinations of 24 students taken 3 at a time.

The formula for combinations is given by:

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

where \( n \) is the total number of items (students in this case), \( r \) is the number of items to choose (3 students), and \( ! \) denotes factorial.

In our case, \( n = 24 \) and \( r = 3 \). Hence, we have:

\[
C(24, 3) = \frac{24!}{3!(24 - 3)!} = \frac{24!}{3! \times 21!}
\]

This simplifies to:

\[
C(24, 3) = \frac{24 \times 23 \times 22}{3 \times 2 \times 1}
\]

Now we calculate the numerator:

\[
24 \times 23 = 552
\]

\[
552 \times 22 = 12144
\]

Now calculate the denominator:

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

Finally, we divide the numerator by the denominator:

\[
C(24, 3) = \frac{12144}{6} = 2024
\]

Thus, the number of ways Mrs. Plum can select three students to sit in the front row is \( \boxed{2024} \).