Question
How many different three-member teams can be formed from six students?(1 point)
Responses
20
20
120
120
216
216
720
Responses
20
20
120
120
216
216
720
Answers
GPT-4o mini
To find the number of different three-member teams that can be formed from six students, we can 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, 6 students), \( r \) is the number of items to choose (in this case, 3 students), and \( ! \) denotes factorial.
Here, \( n = 6 \) and \( r = 3 \). Plugging these values into the formula, we get:
\[
\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!}
\]
Calculating the factorials:
- \( 6! = 720 \)
- \( 3! = 6 \)
Now, substitute these values into the equation:
\[
\binom{6}{3} = \frac{720}{6 \cdot 6} = \frac{720}{36} = 20
\]
Thus, the number of different three-member teams that can be formed from six students is **20**.
\[
\binom{n}{r} = \frac{n!}{r!(n-r)!}
\]
where \( n \) is the total number of items to choose from (in this case, 6 students), \( r \) is the number of items to choose (in this case, 3 students), and \( ! \) denotes factorial.
Here, \( n = 6 \) and \( r = 3 \). Plugging these values into the formula, we get:
\[
\binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3! \cdot 3!}
\]
Calculating the factorials:
- \( 6! = 720 \)
- \( 3! = 6 \)
Now, substitute these values into the equation:
\[
\binom{6}{3} = \frac{720}{6 \cdot 6} = \frac{720}{36} = 20
\]
Thus, the number of different three-member teams that can be formed from six students is **20**.