Question

of 15 possible books, you plan to take 3 with you on vacation. how many different collections of 3 books can you take?
you can take □ different collections of 3 books on vacation with you.

Explanation:

Step1: Identify combination formula

We use combinations since order doesn't matter for collections. The formula for combinations is:
$$C(n,k)=\frac{n!}{k!(n-k)!}$$
where $n=15$ (total books), $k=3$ (books to take).

Step2: Substitute values into formula

$$C(15,3)=\frac{15!}{3!(15-3)!}=\frac{15!}{3! \times 12!}$$

Step3: Simplify the factorial expression

Cancel $12!$ from numerator and denominator:
$$C(15,3)=\frac{15 \times 14 \times 13}{3 \times 2 \times 1}$$

Step4: Calculate the final value

$$\frac{15 \times 14 \times 13}{6}=\frac{2730}{6}=455$$

Answer:

455