The problem of calculating 9C3 times 26P3 can be a complex one, but with the right approach, it can be solved efficiently. To start, let's break down what each part of the expression means. 9C3 refers to the combination of 9 items taken 3 at a time, which is calculated as 9! / (3!(9-3)!), where "!" denotes factorial, the product of an integer and all the integers below it. On the other hand, 26P3 refers to the permutation of 26 items taken 3 at a time, calculated as 26! / (26-3)!. Understanding these definitions is crucial for solving the problem.
Calculating 9C3
To calculate 9C3, we apply the formula for combinations: 9C3 = 9! / (3! * (9-3)!). Calculating the factorials: 9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 362,880, 3! = 3 * 2 * 1 = 6, and (9-3)! = 6! = 6 * 5 * 4 * 3 * 2 * 1 = 720. Plugging these into the formula gives us 9C3 = 362,880 / (6 * 720) = 362,880 / 4,320 = 84.
Calculating 26P3
For 26P3, we use the permutation formula: 26P3 = 26! / (26-3)!. Since (26-3)! = 23!, we only need the first three factors of 26! because the rest cancels out with 23!. So, 26P3 = (26 * 25 * 24 * 23!) / 23! = 26 * 25 * 24 = 15,600.
Now, to find the product of 9C3 and 26P3, we multiply the results of each: 84 * 15,600 = 1,310,400. This calculation demonstrates a straightforward approach to solving the given problem, leveraging the definitions and formulas for combinations and permutations.
| Calculation | Result |
|---|---|
| 9C3 | 84 |
| 26P3 | 15,600 |
| 9C3 * 26P3 | 1,310,400 |
Understanding Combinations and Permutations
Combinations, denoted as nCr, represent the number of ways to choose r items from a set of n items without regard to the order. The formula for combinations is n! / (r!(n-r)!), where “!” denotes factorial. Permutations, denoted as nPr, represent the number of ways to choose r items from a set of n items with regard to the order, calculated as n! / (n-r)!. The distinction between these two is critical in solving problems that involve selecting or arranging items.
Applications of Combinations and Permutations
These mathematical concepts have numerous applications in real-world scenarios, including statistics, probability, computer science, and more. For instance, combinations are used in statistics to calculate probabilities, while permutations are essential in computer science for algorithms that involve arranging data. Understanding and being able to apply these concepts can provide powerful tools for problem-solving across various disciplines.
In the context of the problem 9C3 times 26P3, recognizing the formulas and calculations involved is key to finding the solution efficiently. It also highlights the importance of mathematical fundamentals in tackling complex problems, demonstrating how breaking down a problem into manageable parts and applying the right mathematical concepts can lead to a straightforward solution.
What is the difference between combinations and permutations?
+Combinations (nCr) are the number of ways to choose r items from n without regard to order, calculated as n! / (r!(n-r)!), while permutations (nPr) consider the order, calculated as n! / (n-r)!. This fundamental difference affects how items are selected and arranged.
How are combinations and permutations applied in real-world scenarios?
+Both have numerous applications. Combinations are crucial in statistics for probability calculations, while permutations are vital in computer science for data arrangement algorithms. They also appear in fields like engineering, economics, and biology, among others, wherever selection or arrangement of items is involved.
