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?

Verified step by step guidance

1. 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.
2. Recall the formula for combinations, which is given by: $\mathrm{nCr}=\frac{n!}{r!\; (n-r)!}$, where $n$ is the total number of items, and $r$ is the number of items to choose.
3. Substitute the given values into the formula: $\frac{17!}{8!\; (17\; -\; 8)!}$ which simplifies to $\frac{}{}$.
4. Calculate the factorial values or use simplification techniques to reduce the factorial expressions before multiplying and dividing to avoid large numbers.
5. Evaluate the simplified expression to find the total number of different groups of 8 children that can be formed from 17 children.

Key Concepts

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 can be formed from a larger set when order does not matter. In this problem, selecting 8 children out of 17 involves combinations because the order of selection is irrelevant, only the group composition matters.