Step-by-Step Instructions
Understand the Problem and Identify Inputs
Identify the total number of items (n) and the number of items to choose (r). Make sure you understand what you are trying to find.
Calculate Factorials
Calculate the factorials of n, r, and n-r by multiplying the numbers in descending order.
Apply the Formula
Plug the factorials into the combinations formula: nCr = n! / (r! * (n-r)!).
Simplify the Expression
Simplify the expression by performing the division and following the order of operations (PEMDAS).
Interpret the Result
Interpret the result, which represents the number of ways to choose r items from a set of n items without replacement.
Introduction to Combinations
Combinations, denoted as nCr, are a way to calculate the number of ways to choose r items from a set of n items without replacement. The order of the items does not matter.
Formula
The formula for combinations is nCr = n! / (r! * (n-r)!), where ! denotes the factorial function (e.g., 5! = 5 * 4 * 3 * 2 * 1).
Step-by-Step Guide
To calculate combinations by hand, follow these steps:
Step 1: Understand the Problem and Identify Inputs
First, identify the total number of items (n) and the number of items to choose (r). Make sure you understand what you are trying to find.
Step 2: Calculate Factorials
Next, calculate the factorials of n, r, and n-r. You can do this by multiplying the numbers in descending order. For example, if n = 5, then n! = 5 * 4 * 3 * 2 * 1 = 120.
Step 3: Apply the Formula
Now, plug the factorials into the combinations formula: nCr = n! / (r! * (n-r)!).
Step 4: Simplify the Expression
Simplify the expression by performing the division. Make sure to follow the order of operations (PEMDAS).
Step 5: Interpret the Result
Finally, interpret the result. The result represents the number of ways to choose r items from a set of n items without replacement.
Worked Example
Suppose we want to calculate the number of ways to choose 3 items from a set of 5 items. Here, n = 5 and r = 3. n! = 5 * 4 * 3 * 2 * 1 = 120 r! = 3 * 2 * 1 = 6 (n-r)! = (5-3)! = 2 * 1 = 2 Now, apply the formula: nCr = 120 / (6 * 2) = 120 / 12 = 10 Therefore, there are 10 ways to choose 3 items from a set of 5 items without replacement.
Common Mistakes to Avoid
- Forgetting to calculate the factorial of (n-r)
- Not following the order of operations (PEMDAS)
- Using the wrong formula (e.g., permutations instead of combinations)
When to Use a Calculator
While it's possible to calculate combinations by hand, it's often more convenient to use a calculator, especially for large values of n and r. Most calculators have a built-in combinations function, or you can use the formula and let the calculator perform the calculations.