# 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.