a dance instructor chose four of his 10 students to be on stage for a performance. if order does not matter…

a dance instructor chose four of his 10 students to be on stage for a performance. if order does not matter, in how many different ways can the instructor choose the four students? 10c4 = 10! / (10 - 4)!4! 210 1,260 6,300 25,200

a dance instructor chose four of his 10 students to be on stage for a performance. if order does not matter, in how many different ways can the instructor choose the four students? 10c4 = 10! / (10 - 4)!4! 210 1,260 6,300 25,200

Answer

Explanation:

Step1: Recall combination formula

The combination formula is ${n}C{r}=\frac{n!}{(n - r)!r!}$, where $n$ is the total number of items, and $r$ is the number of items to be chosen. Here $n = 10$ and $r=4$.

Step2: Calculate factorial values

$n!=n\times(n - 1)\times\cdots\times1$. So, $10! = 10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $(10 - 4)!=6!=6\times5\times4\times3\times2\times1$ and $4!=4\times3\times2\times1$. Then ${10}C{4}=\frac{10!}{6!4!}=\frac{10\times9\times8\times7\times6!}{6!\times4\times3\times2\times1}$.

Step3: Simplify the expression

Cancel out the $6!$ terms. We get $\frac{10\times9\times8\times7}{4\times3\times2\times1}=\frac{5040}{24}=210$.

Answer:

210