Пошаговые инструкции
Define Your Variables
First, identify the total number of items (n) and the number of items to choose (r). For example, if you want to choose 3 items from a set of 5, then n = 5 and r = 3.
Calculate the Factorial of n
Next, calculate the factorial of n. Using the example above, calculate 5! = 5*4*3*2*1 = 120.
Calculate the Factorial of r
Then, calculate the factorial of r. Using the example above, calculate 3! = 3*2*1 = 6.
Calculate the Factorial of n-r
After that, calculate the factorial of n-r. Using the example above, calculate 5-3 = 2, then 2! = 2*1 = 2.
Apply the Formula
Finally, plug in the values into the formula: nCr = n! / (r!(n-r)!). Using the example above, 5C3 = 120 / (6*2) = 120 / 12 = 10.
Verify with Pascal's Triangle (Optional)
As an optional step, you can verify your result using Pascal's triangle. The row corresponding to n will have the value of nCr at the rth position (starting from 0). Using the example above, the 5th row of Pascal's triangle is 1 5 10 10 5 1, and the 3rd position (starting from 0) is indeed 10.
Introduction to Combinations
Combinations, denoted as nCr, are a fundamental concept in mathematics and statistics. The formula for calculating combinations is nCr = n! / (r!(n-r)!), where n is the total number of items, r is the number of items to choose, and ! denotes the factorial function. In this guide, we will walk you through the steps to calculate combinations manually.
Understanding the Formula
The formula for combinations is nCr = n! / (r!(n-r)!). This formula can be broken down into three parts: calculating the factorial of n, calculating the factorial of r, and calculating the factorial of n-r.
Factorial Function
The factorial function is denoted by the exclamation mark (!) and is calculated by multiplying all positive integers up to that number. For example, 5! = 54321 = 120.
Step-by-Step Calculation
To calculate combinations, follow these steps: