Problem 64 - Algebra and Trigonometry Real Mathematics, Real People

Question

You can answer any 18 questions from a total of 20 questions on an exam. In how many different ways can you select the questions?

Step-by-Step Solution

Answer

There are 190 different ways to select the questions.

Step 1: Identify the parameters for the combination formula

Here, the total number of items to choose from \(n\) is 20 (the total number of questions), and the number of items to choose \(k\) is 18 (the total questions we can answer).

Step 2: Substitute the values into the combination formula

Substitute \(n = 20\) and \(k = 18\) into the combination formula \[C(n,k) = \frac{ n! }{ k!(n-k)! }\]. So, it becomes \[C(20,18) = \frac{ 20! }{ 18!(20-18)! }\].

Step 3: Calculate the factorial of the numbers and simplify

Calculate note that \(20! = 20 \times 19 \times 18! \) and \(2! = 2\). Substituting these values into the equation \[C(20,18) = \frac{ 20 \times 19 \times 18! }{ 18! \times 2}\], the \(18!\) in the numerator and the denominator cancel out, giving \(C(20,18) = \frac{ 20 \times 19}{2} = 190.\]

Key Concepts

Combination Formula

When solving problems within combinatorics that involve selecting a certain number of items from a larger set, we use the concept of combinations. In these scenarios, the order in which items are selected does not matter---this is a key distinction from permutations. To calculate the number of combinations, we use the combination formula:

• \(C(n, k) = \frac{n!}{k!(n-k)!}\)

Here, \(n\) is the total number of items from which we can choose, and \(k\) is the number of items we want to select. This formula gives us the total number of ways to choose \(k\) items from \(n\) without regard to order.
In the context of the exam problem, choosing 18 questions out of 20, the combination formula helps us to find how many ways we can select which 18 questions to answer. As the solution demonstrated, substituting in \(n = 20\) and \(k = 18\) into the formula leads us to solving a manageable arithmetic problem once we cancel the common terms.

Factorial Calculation

Factorials play a vital role in combinatorics and are central to calculating both permutations and combinations. A factorial, denoted by an exclamation mark (!), is the product of all positive integers up to a certain number. For example, \(n! = n \times (n-1) \times (n-2) \times \ldots \times 1\).

• For 5!, the calculation would be \(5 \times 4 \times 3 \times 2 \times 1 = 120\)

In our exercise, the use of factorials simplifies the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\) as it allows common factors in the numerator and denominator to cancel each other out, making the computation of combinations easier. Understanding how to compute \(20!\), which expands to \(20 \times 19 \times 18!\), and simplifying by canceling out \(18!\) with the factorial in the denominator illustrates an efficient strategy for determining combinations without manually calculating enormous products.

Permutations and Combinations

Understanding the difference between permutations and combinations is crucial in combinatorics. While both involve selection from a set, their uses and computations differ depending on whether order matters.

Permutations

Permutations are used when the order of selection does matter. For example, arranging books on a shelf. The permutation formula is \(P(n, k) = \frac{n!}{(n-k)!}\), which counts how many ways \(k\) items can be ordered from a total of \(n\) items.

• For instance, if you want to arrange 3 books from a total of 5, permutations would count every possible sequence.

Combinations

Combinations ignore the order of selection. This is why the combination formula \(C(n, k) = \frac{n!}{k!(n-k)!}\) includes division by \(k!\), to remove the effect of ordering the items.

• In our question, choosing 18 questions out of 20, combinations count all possible sets of 18, not caring about which question comes first.

Understanding these differences helps to decide which approach and formula to use when faced with a problem, guiding whether order is relevant or should be considered.