combination

Combination

Today, we will learn about combinations. Combinations are a way to select items from a group, where the order of selection does not matter. This is different from permutations, where the order does matter. We will explore what combinations are, how to calculate them, and see some examples from everyday life.

Understanding Combinations

In mathematics, a combination is a selection of items from a larger pool, where the order of the items does not matter. For example, if you have a basket of 3 fruits: an apple, a banana, and a cherry, and you want to pick 2 fruits, the combinations would be:

• Apple and Banana
• Apple and Cherry
• Banana and Cherry

Notice that "Apple and Banana" is the same as "Banana and Apple" because the order does not matter in combinations.

Formula for Combinations

To calculate the number of combinations, we use the combination formula. The formula to find the number of ways to choose \( r \) items from \( n \) items is:

\( C(n, r) = \frac{n!}{r!(n-r)!} \)

Here, \( n! \) (n factorial) means the product of all positive integers up to \( n \). For example, \( 4! = 4 \times 3 \times 2 \times 1 = 24 \).

Example 1: Choosing Fruits

Let's say you have 4 fruits: an apple, a banana, a cherry, and a date. You want to choose 2 fruits. How many combinations are there?

Using the formula:

\( C(4, 2) = \frac{4!}{2!(4-2)!} = \frac{4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} = \frac{24}{4} = 6 \)

So, there are 6 combinations: Apple and Banana, Apple and Cherry, Apple and Date, Banana and Cherry, Banana and Date, Cherry and Date.

Example 2: Choosing Team Members

Imagine you have 5 friends: Alice, Bob, Charlie, David, and Eve. You want to form a team of 3 members. How many different teams can you form?

Using the formula:

\( C(5, 3) = \frac{5!}{3!(5-3)!} = \frac{5 \times 4 \times 3 \times 2 \times 1}{3 \times 2 \times 1 \times 2 \times 1} = \frac{120}{12} = 10 \)

So, there are 10 different teams you can form.

Example 3: Choosing Books

Suppose you have 6 books and you want to choose 4 to take on a trip. How many different groups of books can you choose?

Using the formula:

\( C(6, 4) = \frac{6!}{4!(6-4)!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{4 \times 3 \times 2 \times 1 \times 2 \times 1} = \frac{720}{48} = 15 \)

So, there are 15 different groups of books you can choose.

Real-World Applications

Combinations are used in many real-world situations. Here are a few examples:

• Lottery: When you choose numbers for a lottery ticket, the order of the numbers does not matter. The winning numbers are a combination.
• Sports Teams: Coaches often need to select players for a team from a larger group. The order in which players are selected does not matter, so this is a combination.
• Menu Choices: When you choose items from a menu, like picking 2 toppings for a pizza from a list of 5, the order of the toppings does not matter. This is a combination.

Summary

Today, we learned about combinations. A combination is a way to select items from a group where the order does not matter. We use the formula \( C(n, r) = \frac{n!}{r!(n-r)!} \) to calculate the number of combinations. We saw examples of choosing fruits, team members, and books. Combinations are used in many real-world situations like lotteries, sports teams, and menu choices.