c(n, r) represents the number of combinations of n items taken r at a time, calculated using the formula $$c(n, r) = \frac{n!}{r!(n-r)!}$$. This term is essential for understanding how to count selections where the order does not matter, distinguishing it from permutations, where order is significant. It is commonly used in probability and statistics to determine how many ways you can choose a subset from a larger set.
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c(n, r) is only defined when n is greater than or equal to r, meaning you can't choose more items than are available.
The value of c(n, r) is symmetric, meaning c(n, r) = c(n, n - r). This shows that choosing r items from n is the same as leaving out n - r items.
For small values of n and r, you can calculate c(n, r) easily by hand; however, for larger values, calculators or software are often used.
c(n, 0) is always 1 since there is exactly one way to choose no items from a set: choose nothing.
c(n, 1) equals n because there are n ways to choose one item from a group of n items.
Review Questions
How do you calculate c(n, r), and what does it represent in practical scenarios?
To calculate c(n, r), you use the formula $$c(n, r) = \frac{n!}{r!(n-r)!}$$. This represents the number of ways to choose r items from a total of n items without regard to the order of selection. In practical terms, this can apply to scenarios such as selecting team members from a larger group or determining possible lottery combinations.
Compare and contrast combinations and permutations using c(n, r) and its related formula.
Combinations (c(n, r)) focus on selecting items where the order does not matter, while permutations (p(n, r)) take order into account. For example, when choosing 3 fruits from a basket of 5, c(5, 3) counts combinations like {apple, banana, cherry}, whereas p(5, 3) would consider different arrangements of those fruits like {apple, banana, cherry} versus {banana, apple, cherry}. The formulas differ significantly: c(n, r) uses factorials to account for combinations while p(n, r) includes additional factors to represent the ordering.
Evaluate how the concept of c(n, r) contributes to understanding probability and its applications in real-world problems.
Understanding c(n, r) is critical for calculating probabilities in scenarios where outcomes can be selected without regard for order. For instance, in a card game where you need to calculate the likelihood of drawing a specific combination of cards from a deck, knowing how many unique combinations exist allows for precise probability calculations. This concept extends to fields like genetics for predicting traits or economics for evaluating market options based on varied selections.
Related terms
Factorial: The product of all positive integers up to a given number n, denoted as n!, which is crucial for calculating combinations and permutations.
Permutations: Different arrangements of a set of items where the order matters, represented as p(n, r) and calculated differently than combinations.
Binomial Coefficient: Another name for c(n, r), often used in the context of binomial expansions and probability.