Question

suppose you choose 4 books to read from a summer reading list of 12 books. how many different combinations of books are possible?

\\{}_{n}c_{r} = \frac{n!}{r!(n-r)!}\\

Explanation:

Identify the given parameters

We need to choose \(r = 4\) books from a total of \(n = 12\) books.

Apply the combination formula

Using the Factorial Notation knowledge point

$$ _{12}C_{4} = \frac{12!}{4!(12-4)!} = \frac{12!}{4! \cdot 8!} $$

Simplify the factorial expression

Expand the larger factorial to cancel terms:

$$ _{12}C_{4} = \frac{12 \times 11 \times 10 \times 9 \times 8!}{4! \cdot 8!} = \frac{12 \times 11 \times 10 \times 9}{4 \times 3 \times 2 \times 1} $$

Calculate the final value

Simplify the fraction:

$$ _{12}C_{4} = \frac{11880}{24} = 495 $$

Answer:

495