Question

How many ways can a dance instructor send 6 of her 9 students to a summer dance program

Answer 1

The question relates to the number of combinations of 9 students taken 6 at a time:
$9C6=\frac{9!}{6!3!}=\frac{9\times 8\times 7}{3\times2\times1}=84\ ways$

Answer 2

Answer:  there are 84 ways to find 6 students from 9 students .

Step-by-step explanation:

Since we have given that

Total number of students to a summer dance program =9

Total number of students who can be send by dance instructor = 6

Number of ways to choose 6 students from 9 students i.e.

$^9C_6$

As we know the "Combination formula " i.e.

$^nC_r=\frac{n!}{(n-r)!r!}$

where n denotes the total number of students

r denotes the selected students

So,

$^9C_r=\frac{9!}{(9-6)!6!}\\\\^9C_r=\frac{9!}{3!\times 6!}\\\\^9C_r=\frac{9\times 8\times 7}{3\times 2\times 1}\\\\^9C_r=3\times 4\times 7\\\\^9C_r=84$

Hence, there are 84 ways to find 6 students from 9 students .