C(n, r)
from class:
College Algebra
Definition
C(n, r), also known as the binomial coefficient, represents the number of ways to choose r items from a set of n items, without regard to order. It is a fundamental concept in probability and combinatorics.
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5 Must Know Facts For Your Next Test
- The formula for C(n, r) is: $C(n, r) = \frac{n!}{r!(n-r)!}$, where n! represents the factorial of n.
- C(n, r) is used to calculate the number of possible outcomes in probability problems involving combinations, such as the number of ways to choose a subset of items from a set.
- The value of C(n, r) is always an integer, and it represents the number of unique subsets of size r that can be formed from a set of n items.
- C(n, r) is a symmetric function, meaning that C(n, r) = C(n, n-r).
- The binomial coefficient C(n, r) is a crucial concept in probability theory, as it is used to calculate the probability of specific outcomes in binomial probability distributions.
Review Questions
- Explain the relationship between C(n, r) and the concept of combinations.
- The binomial coefficient C(n, r) represents the number of ways to choose r items from a set of n items, without regard to order. This is directly related to the concept of combinations, which is the number of unique subsets of size r that can be formed from a set of n items. The formula for C(n, r) reflects this relationship, as it takes into account the factorial terms to account for the number of possible arrangements within each subset.
- Describe how C(n, r) is used in the context of probability problems.
- In probability theory, C(n, r) is a crucial concept for calculating the probability of specific outcomes, particularly in binomial probability distributions. The binomial coefficient is used to determine the number of possible ways a certain event can occur, which is then used to calculate the overall probability of that event happening. For example, if you are rolling a dice 5 times and want to know the probability of getting exactly 3 sixes, you would use C(5, 3) to calculate the number of ways to get 3 sixes out of 5 rolls, and then divide that by the total number of possible outcomes.
- Analyze the properties of C(n, r) and explain how they can be used to simplify calculations.
- The binomial coefficient C(n, r) has several important properties that can be leveraged to simplify calculations. First, the formula $C(n, r) = \frac{n!}{r!(n-r)!}$ shows that the value of C(n, r) is always an integer, which can be useful when working with probability problems. Additionally, the fact that C(n, r) is a symmetric function, meaning that C(n, r) = C(n, n-r), allows you to choose the smaller of the two values when calculating the coefficient. This can be particularly helpful when n is much larger than r. Finally, the recursive nature of the formula, where C(n, r) = C(n-1, r-1) + C(n-1, r), can be used to build up values of C(n, r) efficiently, especially for larger values of n and r.
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