Combination Calculator
Welcome to the Combination Calculator. This tool allows you to calculate the number of combinations for a given set of items. A combination is a collection of objects in any sequence. This is often used in probability and statistics.
Result :
How to Use the Combination Calculator
- Enter the Value of n (Total Number of Items)
Locate the input field labeled "Enter n value." Enter the total number of items or elements in this field. This is the number from which you are selecting items.
- Enter the Value of r (Number of Items to Select)
Locate the input field labeled "Enter k value." Enter the number of items you want to select from the total number of items.
- Calculate the Combination
After entering the values for n and k, click the "Calculate" button. The calculator will process the input and display the result, showing the number of combinations possible.
- Review the Detailed Solution
The calculator will provide a step-by-step solution using the combination formula \( \mathrm{ \mbox{nCk} = \frac{n!}{r!(n-k)!} } \). Review the detailed steps to understand how the result was computed.
Combination Formula
The combination formula is used to determine the number of possible selections in a set when the order of the elements does not matter. The formula is given by:
\( \mathrm{ \mbox{nCr} = \frac{n!}{r!(n-r)!} } \)
Where:
- n = Total number of items
- r = Number of items to select
This formula calculates the number of ways to choose r items from n items without considering the order of selection.
How to Calculate Combinations Manually
To calculate combinations manually, follow these steps:
- Step 1: Write down the combination formula: \( \mathrm{ \mbox{nCr} = \frac{n!}{r!(n-r)!} } \).
- Step 2: Calculate the factorial of n, which is the product of all positive integers up to n.
- Step 3: Calculate the factorial of r and the factorial of (n - r).
- Step 4: Substitute the factorial values into the formula.
- Step 5: Simplify the expression to find the number of combinations.
Example: To calculate \( \mathrm{ \mbox{15C10} } \):
- Calculate \( 15! \)
- Calculate \( 10! \) and \( 5! \) (since \( 15 - 10 = 5 \))
- Substitute into the formula: \( \mathrm{ \mbox{15C10} = \frac{15!}{10!5!} } \)
- Simplify the expression to get the result.