if the math olympiad club consists of 12 stud...
if the math olympiad club consists of 12 students, how many different teams of 3 students can be formed for competitions?
Answer
# Explanation:
## Step1: Identify the combination formula
The formula for combinations is $C(n,r)=\frac{n!}{r!(n - r)!}$, where $n$ is the total number of items, and $r$ is the number of items to be chosen. Here, $n = 12$ and $r=3$.
## Step2: Calculate factorial values
$n!=12! = 12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $r!=3!=3\times2\times1$, and $(n - r)!=(12 - 3)!=9!=9\times8\times7\times6\times5\times4\times3\times2\times1$. Then $C(12,3)=\frac{12!}{3!(12 - 3)!}=\frac{12\times11\times10\times9!}{3\times2\times1\times9!}$.
## Step3: Simplify the expression
Cancel out the $9!$ terms. We get $C(12,3)=\frac{12\times11\times10}{3\times2\times1}=\frac{1320}{6}=220$.
# Answer:
220