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?\n$_{10}c_{4}=\frac{10!}{(10 - 4)!4!}$\no 210\no 1,260\no 6,300\no 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?\n$_{10}c_{4}=\frac{10!}{(10 - 4)!4!}$\no 210\no 1,260\no 6,300\no 25,200

Answer

Explanation:

Step1: Recall combination formula

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

Step2: Calculate factorial values

$n!=10! = 10\times9\times8\times7\times6\times5\times4\times3\times2\times1$, $(n - r)!=(10 - 4)!=6!=6\times5\times4\times3\times2\times1$, $r!=4!=4\times3\times2\times1$. 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:

A. 210