Question

three out of seven students in the cafeteria line are chosen to answer survey questions.
how many different combinations of three students are possible?
$_{7}c_{3} = \frac{7!}{(7-3)!3!}$
○ 7
○ 35
○ 70
○ 210

Explanation:

Step1: Recall combination formula

The combination formula is $_nC_r = \frac{n!}{(n-r)!r!}$, where $n=7$, $r=3$.

Step2: Substitute values into formula

$$_7C_3 = \frac{7!}{(7-3)!3!} = \frac{7!}{4!3!}$$

Step3: Expand factorials and simplify

$$\frac{7\times6\times5\times4!}{4!\times3\times2\times1} = \frac{7\times6\times5}{3\times2\times1}$$

Step4: Calculate the final value

$$\frac{210}{6} = 35$$

Answer:

35 (Option B)