Use the formula for nCr to solve Exercises 49–56. Of 12 possible books, you plan to take 4 with you on vacation. How many different collections of 4 books can you take?

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Identify the problem as a combination problem where order does not matter. You want to find the number of ways to choose 4 books out of 12 without regard to order.

Recall the formula for combinations, which is given by $nCr = \frac{n!}{r!(n-r)!}$, where $n$ is the total number of items, and $r$ is the number of items to choose.

Substitute the given values into the formula: $n = 12$ and $r = 4$, so the expression becomes $\frac{12!}{4!(12-4)!}$.

Simplify the factorial expressions in the numerator and denominator to make the calculation easier. For example, expand \$12!$ as \(12 \times 11 \times 10 \times 9 \times 8!$ and cancel the \)8!$ in numerator and denominator.

Calculate the remaining multiplication and division to find the number of different collections of 4 books you can take.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Combination Formula (nCr)

The combination formula, denoted as nCr, calculates the number of ways to choose r items from a set of n distinct items without regard to order. It is given by nCr = n! / [r! (n - r)!], where '!' denotes factorial. This formula is essential for counting selections where order does not matter.

Factorials

A factorial, represented by n!, is the product of all positive integers from 1 up to n. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in permutations and combinations to calculate the total number of arrangements or selections.

Counting Without Replacement

When selecting items from a set without replacement, each chosen item is not returned to the set, reducing the total number available for subsequent choices. Combinations apply here because the order of selection does not matter, only the group chosen.