Question

Your soccer team has 20 players. How many groups of 5 players can be formed?

Question 17 options:

54,000


15,504


25,000


3,200,000

Answer 1

To determine how many groups of 5 players can be formed from a soccer team of 20 players, we can use the combination formula, which is given by:

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

where \( n \) is the total number of items (in this case, players), \( r \) is the number of items to choose (in this case, 5), and \( ! \) denotes factorial.

For this problem, \( n = 20 \) and \( r = 5 \):

\[
C(20, 5) = \frac{20!}{5! \cdot (20-5)!} = \frac{20!}{5! \cdot 15!}
\]

We can simplify this calculation as follows:

\[
C(20, 5) = \frac{20 \times 19 \times 18 \times 17 \times 16}{5 \times 4 \times 3 \times 2 \times 1}
\]

Calculating the numerator:

\[
20 \times 19 = 380
\]
\[
380 \times 18 = 6840
\]
\[
6840 \times 17 = 116280
\]
\[
116280 \times 16 = 1860480
\]

The numerator is 1,860,480.

Now, calculating the denominator:

\[
5 \times 4 \times 3 \times 2 \times 1 = 120
\]

Now, we can divide the numerator by the denominator:

\[
C(20, 5) = \frac{1860480}{120} = 15504
\]

Thus, the number of groups of 5 players that can be formed is **15,504**.

So, the answer is:

**15,504**