Question

find the number of different ways that an instructor can choose 7 students from a class of 12 students for a field trip.

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 $r=7$.

Step2: Calculate factorial values

We know that $n!=n\times(n - 1)\times\cdots\times1$. So, $12! = 12\times11\times10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $7! = 7\times6\times5\times4\times3\times2\times1$, and $(12 - 7)!=5!=5\times4\times3\times2\times1$. Then $C(12,7)=\frac{12!}{7!(12 - 7)!}=\frac{12!}{7!5!}=\frac{12\times11\times10\times9\times8\times7!}{7!\times5\times4\times3\times2\times1}$.

Step3: Simplify the expression

The $7!$ in the numerator and denominator cancels out. We have $\frac{12\times11\times10\times9\times8}{5\times4\times3\times2\times1}=\frac{95040}{120}=792$.

Answer:

792