Problem 19 - A Graphical Approach to College Algebra

Question

Evaluate each expression. $$C(4,2)$$

Step-by-Step Solution

Answer

There are 6 ways to choose 2 items from 4.

Step 1: Understand the Combination Formula

The combination formula, denoted as \( C(n, r) \), is used to determine how many ways you can choose \( r \) objects from a set of \( n \) objects, without regard to the order of selection. The formula is given by: \[ C(n, r) = \frac{n!}{r!(n-r)!} \]where \( n! \) (n factorial) is the product of all positive integers up to \( n \).

Step 2: Plug in the Values for n and r

For the expression \( C(4, 2) \), we have \( n = 4 \) and \( r = 2 \). Substitute these values into the combination formula: \[ C(4, 2) = \frac{4!}{2!(4-2)!} \]

Step 3: Calculate Factorials

Compute the factorial for each necessary component: - \( 4! = 4 \times 3 \times 2 \times 1 = 24 \)- \( 2! = 2 \times 1 = 2 \)- \( (4-2)! = 2! = 2 \times 1 = 2 \)

Step 4: Substitute Factorials into Formula

Now that we have calculated the factorials, substitute them back into the combination formula:\[ C(4, 2) = \frac{24}{2 \times 2} = \frac{24}{4} \]

Step 5: Perform the Division

Divide the numbers:\[ \frac{24}{4} = 6 \]

Key Concepts

Combination formula

Combinations are a way to choose items from a group, where the order of selection doesn't matter. The combination formula is key in calculating this. It's denoted as \( C(n, r) \) and it calculates the number of ways to select \( r \) items from \( n \) total items. The formula is:

• \( C(n, r) = \frac{n!}{r!(n-r)!} \)

The exclamation mark \(!\) represents a factorial, which we'll explore in the next section. To better understand, let's say you want to pick 2 fruits from a bowl of 4 different fruits. Whether you pick apple then banana, or banana then apple, the outcome is the same. So, you use combinations to find these possibilities. Plug the numbers into the formula: for \( C(4,2) \), it means picking 2 items from 4, and it becomes \( \frac{4!}{2!(4-2)!} \). You solve each part step by step.

Factorials

Factorials are a fundamental part of understanding combinations and permutations. A factorial, represented by \( n! \), is the product of all positive integers up to that number \( n \). For instance, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \). Factorials simplify expressions in combination calculations.
Understanding how to compute factorials makes solving problems easier.

• \( 0! \) is defined as \( 1 \).
• Factorials grow fast. For example, \( 5! = 120 \).

Factorials are useful in many areas of mathematics beyond combinations, like permutations and sequences. In our combination example \( C(4,2) \), you calculate each factorial separately and substitute back into the formula to find the answer. This leads to a more straightforward calculation and eventually, a clear answer.

Permutations

Permutations are similar to combinations, but with a crucial difference: the order of selection matters. While combinations focus on selection without concern for order, permutations consider different sequences as different outcomes.
For instance, if you’re seating 4 people, the order in which they sit makes a new permutation. The formula for permutations is:

• \( P(n, r) = \frac{n!}{(n-r)!} \)

This shows how many ways to arrange \( r \) items from \( n \).

• A simple example is finding the number of ways to arrange 3 books on a shelf from 5, this is \( P(5,3) \) and equals \( \frac{5!}{(5-3)!} \).

The concept is closely linked with combinations, sharing the factorial foundation. However, focus differs: permutations deal extensively with arrangement and sequencing.