Mastering Combination Calculations
Understanding Combinations
Combinations represent the number of distinct groups that can be formed by selecting elements from a larger set without considering the order in which they are chosen. Unlike permutations, where order matters, combinations focus solely on the selection of items. For example, consider selecting a team of three players from a pool of ten. The order in which the players are selected doesn't matter, making it a combination problem.
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Step-by-Step Guide to Calculating Combinations:
1. Identify \( n \) and \( r \):
Determine the total number of items in the set (\( n \)) and the number of items to be selected (\( r \)).
2. Calculate Factorials:
Compute the factorials of \( n \), \( r \), and \( (n - r) \) using the factorial formula.
3. Apply the Combination Formula:
Substitute the values of \( n \) and \( r \) into the combination formula and perform the necessary calculations.
4. Simplify and Solve:
Simplify the expression and perform the division to obtain the final result, which represents the number of combinations.
Example: Calculating Combinations
Suppose you want to calculate the number of ways to choose two books from a shelf of seven books. Using the combination formula:
\[ \binom{7}{2} = \frac{7!}{2! \cdot (7 - 2)!} \]
\[ = \frac{7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{(2 \times 1) \times (5 \times 4 \times 3 \times 2 \times 1)} \]
\[ = \frac{7 \times 6}{2 \times 1} \]
\[ = \frac{42}{2} \]
\[ = 21 \]
Therefore, there are 21 different combinations of two books that can be selected from the shelf of seven books.
Summary
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