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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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.