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Struggling with combination and permutation calculations? You’re not alone! Understanding how to calculate expressions like 9C3 × 26P3 can be challenging but incredibly rewarding. Whether you’re preparing for exams, solving puzzles, or working on real-world problems, mastering these concepts is essential. In this guide, we’ll break down the steps to calculate 9C3 × 26P3 in a simple, easy-to-follow manner. By the end, you’ll have the confidence to tackle similar problems effortlessly. Let’s dive in! (combination and permutation,math problem-solving,calculation guide)
Understanding the Basics: Combinations and Permutations
Before we jump into calculating 9C3 × 26P3, let’s revisit the fundamentals of combinations and permutations. These are two key concepts in combinatorics, a branch of mathematics dealing with counting and arranging objects.
- Combination (nCr): Represents the number of ways to choose r items from a set of n items without regard to order. The formula is:
[ nCr = \frac{n!}{r!(n-r)!} ] - Permutation (nPr): Represents the number of ways to arrange r items from a set of n items where order matters. The formula is:
[ nPr = \frac{n!}{(n-r)!} ]
📌 Note: Factorial (!) means multiplying all whole numbers from that number down to 1. For example, 5! = 5 × 4 × 3 × 2 × 1 = 120.
Step-by-Step Calculation of 9C3
Let’s start by calculating 9C3, which means choosing 3 items from a set of 9 items without considering the order.
- Apply the combination formula:
[ 9C3 = \frac{9!}{3!(9-3)!} = \frac{9!}{3!6!} ] - Calculate the factorials:
[ 9! = 362,880, \quad 3! = 6, \quad 6! = 720 ] - Simplify the expression:
[ 9C3 = \frac{362,880}{6 \times 720} = \frac{362,880}{4,320} = 84 ]
So, 9C3 = 84. (combination formula,factorial calculation,step-by-step math)
Step-by-Step Calculation of 26P3
Next, let’s calculate 26P3, which means arranging 3 items from a set of 26 items where order matters.
- Apply the permutation formula:
[ 26P3 = \frac{26!}{(26-3)!} = \frac{26!}{23!} ] - Simplify the factorials:
[ 26P3 = 26 \times 25 \times 24 ] - Perform the multiplication:
[ 26 \times 25 = 650, \quad 650 \times 24 = 15,600 ]
So, 26P3 = 15,600. (permutation formula,factorial simplification,math tutorial)
Multiplying 9C3 and 26P3
Now that we have both values, let’s multiply them together: 9C3 × 26P3.
- Multiply the results:
[ 84 \times 15,600 = 1,305,600 ]
Therefore, 9C3 × 26P3 = 1,305,600. (multiplication steps,final calculation,math guide)
Summary and Checklist
Here’s a quick recap of the steps we covered:
| Step | Action | Result |
|---|---|---|
| 1 | Calculate 9C3 | 84 |
| 2 | Calculate 26P3 | 15,600 |
| 3 | Multiply the results | 1,305,600 |
Checklist for solving similar problems:
- Identify whether the problem requires combinations, permutations, or both.
- Apply the correct formula for each scenario.
- Calculate factorials carefully.
- Multiply or add results as needed.
With this guide, you’re now equipped to tackle 9C3 × 26P3 and similar problems with ease. Practice makes perfect, so keep exploring and applying these concepts! (summary checklist,math problem checklist,calculation tips)
What is the difference between combination and permutation?
+Combinations focus on selecting items without regard to order, while permutations consider the arrangement of items where order matters.
How do I calculate factorials easily?
+Factorials can be calculated by multiplying all whole numbers from the given number down to 1. Use a calculator for larger numbers.
Can I use these formulas in real-life applications?
+Yes! Combinations and permutations are used in various fields like statistics, cryptography, and even lottery calculations.
