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! }

Explanation:

Step1: Simplify the factorial expressions

$$\begin{align*} _7C_3&=\frac{7!}{(7 - 3)!3!}\\ &=\frac{7!}{4!3!}\\ &=\frac{7\times6\times5\times4!}{4!\times3\times2\times1} \end{align*}$$

Step2: Cancel out the common terms

Cancel out the \(4!\) terms:

$$\begin{align*} _7C_3&=\frac{7\times6\times5}{3\times2\times1}\\ &=\frac{210}{6} \end{align*}$$

Step3: Calculate the final result

$$ _7C_3 = 35 $$

Answer:

35