Question

a science fair has 10 projects, and 3 will be chosen to receive awards. how many different ways can the award recipients be selected?
$_{10}c_3 = \boxed{?}$
hint:
$_{n}c_{r}=\frac{_{n}p_{r}}{r!}$

Explanation:

Step1: Recall combination formula

Combination formula: $${n \choose r} = \frac{n!}{r!(n-r)!}$$ where $n=10$, $r=3$.

Step2: Calculate factorial values

$10! = 10×9×8×7!$, $3! = 6$, $(10-3)! = 7!$.

Step3: Simplify the formula

$${10 \choose 3} = \frac{10×9×8×7!}{3!×7!} = \frac{10×9×8}{6}$$

Step4: Compute the result

$\frac{720}{6} = 120$.

Answer:

120