Use the formula for nCr to solve Exercises 49–56. You volunteer to help drive children at a charity event to the zoo, but you can fit only 8 of the 17 children present in your van. How many different groups of 8 children can you drive?

검증된 단계별 안내

Identify the problem as a combination problem because the order in which the children are chosen does not matter. We want to find the number of ways to choose 8 children out of 17.

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

Substitute the given values into the formula: $n = 17$ and $r = 8$, so the expression becomes $\displaystyle \binom{17}{8} = \frac{17!}{8!(17-8)!} = \frac{17!}{8!9!}$.

Calculate the factorial values or simplify the expression by expanding the factorials partially to make the calculation easier, for example, expand \$17!$ down to \$9!$ to cancel terms.

Evaluate the simplified expression to find the total number of different groups of 8 children that can be formed from 17 children.

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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 groups or subsets 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.

Application of Combinations in Real-Life Problems

Combinations are used to determine how many different groups or selections can be made from a larger set when order does not matter. In this problem, selecting 8 children out of 17 to fit in a van is a practical example where combinations help find the number of possible groups.