Solve: If 12C3 = 220, Find the Value - A Simple Guide
Struggling with combinations problems? Don't worry, you're not alone! Many find them tricky at first. This post will guide you through how to solve problems like "If 12C3 = 220, find the value," breaking down the steps in an easy-to-understand way. Let's dive in!
Understanding Combinations (nCr)
Before we get started, let's understand what combinations are. In math, combinations help us figure out how many ways we can choose items from a larger group, without considering the order in which we pick them. So, choosing apples, bananas, and oranges is the same as choosing oranges, apples, and bananas in a combination.
nCr (or sometimes written as nCr) is the notation used to represent the number of combinations. "n" is the total number of items to choose from, and "r" is the number of items you're choosing.
The formula for combinations is:
nCr = n! / (r! * (n-r)!)
Let's clarify the problem: If 12C3 = 220, find the value. The question is asking us to confirm the result and understand how it is calculated.
Step-by-Step Solution of 12C3
Now, let's apply the formula to solve 12C3:
- Identify n and r: In our problem, n = 12 (the total number of items) and r = 3 (the number of items we're choosing).
- Apply the Formula: Substitute these values into the combinations formula:
12C3 = 12! / (3! * (12-3)!)
- Simplify the Factorials: Let's expand the factorials (remember, a factorial like 5! means 5 * 4 * 3 * 2 * 1):
12! = 12 * 11 * 10 * 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
3! = 3 * 2 * 1 = 6
(12-3)! = 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
12C3 = (12 * 11 * 10 * 9!)/(6 * 9!)
12C3 = (12 * 11 * 10) / 6 - Calculate the Result: Now, let's do the math:
12C3 = (12 * 11 * 10) / 6 = 1320 / 6 = 220
So, 12C3 = 220!
Verification and Understanding
Confirming the Answer: Our calculation of 12C3 = 220 matches the value given in the problem. Great!
What Does it Mean? In simple terms, 12C3 = 220 means that there are 220 different ways to choose a group of 3 items from a set of 12 items, where the order doesn't matter. For example, imagine you have 12 different fruits and you want to pick 3 to make a smoothie. There are 220 possible combinations of fruit you can choose.
Conclusion
We've successfully solved the combinations problem "If 12C3 = 220, find the value" by using the combinations formula and walking through each step.
Combinations are a fundamental concept in probability and statistics. Understanding how to calculate them opens the door to solving many more interesting problems.
Keep practicing! The more you practice, the easier it will become. If you have any questions or want to discuss other combination problems, feel free to leave a comment below!
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