C(n,r) represents the number of ways to choose 'r' items from a total of 'n' items without regard to the order of selection. This is known as a combination and is crucial for calculating probabilities in situations where the arrangement of selected items does not matter. Understanding C(n,r) allows for deeper insights into various probability problems, especially when distinguishing between combinations and permutations.
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C(n,r) is calculated using the formula: $$C(n,r) = \frac{n!}{r!(n-r)!}$$, where '!' denotes factorial.
C(n,r) is symmetric, meaning that C(n,r) = C(n,n-r); choosing r items from n is the same as leaving out n-r items.
The value of C(n,r) is always a whole number and represents a count of distinct groups.
When r = 0 or r = n, C(n,r) equals 1, indicating that there is exactly one way to choose all or none of the items.
C(n,r) becomes particularly useful in probability problems involving random selections and events.
Review Questions
How does C(n,r) differ from permutations, and why is this distinction important in probability calculations?
C(n,r) focuses on the selection of 'r' items from 'n' without considering the order, while permutations take into account the order of items. This distinction is important because many real-world scenarios require counting combinations rather than arrangements. For example, when forming a committee from a group, the order in which members are chosen does not matter, making C(n,r) the appropriate calculation to use for finding probabilities associated with selecting groups.
Explain how you would calculate C(5,2) and what this value represents in a practical scenario.
To calculate C(5,2), use the formula $$C(5,2) = \frac{5!}{2!(5-2)!} = \frac{5!}{2!3!}$$. This simplifies to $$\frac{5 \times 4}{2 \times 1} = 10$$. This value represents the number of ways to choose 2 items from a set of 5 distinct items. For instance, if you have 5 different fruits and want to know how many ways you can select 2 fruits to make a smoothie, C(5,2) gives you the answer.
Evaluate how understanding C(n,r) can enhance your ability to solve complex probability problems involving multiple events.
Understanding C(n,r) enables you to tackle complex probability problems by allowing you to accurately count the different ways outcomes can occur without getting lost in the details of arrangement. For example, in card games where you might draw several cards without replacement, knowing how to apply combinations helps you determine probabilities based on various hands possible. As events become more complicated with different outcomes and combinations required for success, leveraging C(n,r) can simplify calculations and yield correct probabilities more efficiently.
A permutation is an arrangement of items in a specific order, contrasting with combinations where order does not matter.
Factorial: The factorial of a number 'n', denoted as n!, is the product of all positive integers from 1 to n, used in calculating combinations and permutations.