Question

A basketball league’s final four playoff starts from a bracket of 16 teams. How many combinations of 4 teams can be chosen from the round of 16 to be in the final four?(1 point)

Answer 1

To determine how many combinations of 4 teams can be chosen from a bracket of 16 teams, you can use the combination formula:

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

where \( n \) is the total number of teams, \( r \) is the number of teams to choose, and \( ! \) denotes factorial.

In this case, \( n = 16 \) and \( r = 4 \):

\[
\binom{16}{4} = \frac{16!}{4!(16-4)!} = \frac{16!}{4! \cdot 12!}
\]

This simplifies to:

\[
\binom{16}{4} = \frac{16 \times 15 \times 14 \times 13}{4 \times 3 \times 2 \times 1}
\]

Calculating the numerator:

\[
16 \times 15 = 240
\]
\[
240 \times 14 = 3360
\]
\[
3360 \times 13 = 43680
\]

Now, calculating the denominator:

\[
4! = 4 \times 3 \times 2 \times 1 = 24
\]

Now, divide the numerator by the denominator:

\[
\frac{43680}{24} = 1820
\]

Therefore, the number of combinations of 4 teams that can be chosen from the 16 teams in the league is:

\[
\boxed{1820}
\]