Problem 53 - Algebra and Trigonometry

Question

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?

Step-by-Step Solution

Answer

There are 24310 different groups of 8 children you can drive from a group of 17 children.

Step 1: Identify the variables

First identify the number of children you're choosing from, \(n\), and the number of children you're choosing, \(k\). In this case, \(n = 17\) and \(k = 8\).

Step 2: Apply the combination formula

Next, apply the values to the formula for combinations. Given by \(C(n, k) = \frac{n!}{k!(n-k)!}\) substituting \(n = 17\) and \(k = 8\), we get: \(C(17, 8) = \frac{17!}{8!(17-8)!}\)

Step 3: Evaluate the factorials

Evaluate 17!, 8! and 9! (17-8). The factorial of a number is the product of all positive integers less than or equal to that number.

Step 4: Plug in the values and compute

After computing the factorials, plug them into the formula and simplify to get the result. You divide the product of integers from 1 to 17 by the product of integers from 1 to 8 and 1 to 9. The result will give you the number of different groups of 8 children that can be selected from 17 children.

Step 5: Interpret the result

The result gotten from step 4 is the number of different ways you can select 8 children from a group of 17 children. This is the solution to the problem.