A combination is a way of selecting a set of items from a larger group, where the order of the items does not matter. Combinations are a fundamental concept in the fields of mathematics, probability, and the binomial theorem.
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The formula for calculating the number of combinations of r items from a set of n items is $\binom{n}{r} = \frac{n!}{r!(n-r)!}$, where $n!$ represents the factorial of $n$.
Combinations are used in the binomial theorem to expand binomial expressions, such as $(a + b)^n$, where the coefficients of the terms are given by the binomial coefficients.
In probability, combinations are used to calculate the number of ways to select a subset of items from a larger set, which is necessary for determining the probability of events.
Combinations are often used in counting problems, where the order of the selected items is not important, such as in the number of ways to choose a committee of r people from a group of n people.
The binomial coefficient $\binom{n}{r}$ can also be interpreted as the number of ways to arrange r items in a set of n items, where the order of the items does not matter.
Review Questions
Explain how combinations are used in the binomial theorem.
The binomial theorem states that $(a + b)^n = \sum_{r=0}^n \binom{n}{r} a^{n-r} b^r$. The coefficients of the terms in this expansion are given by the binomial coefficients $\binom{n}{r}$, which represent the number of ways to choose r items from a set of n items, where the order does not matter. These binomial coefficients are crucial for the binomial theorem, as they determine the number of terms in the expansion and the coefficients of those terms.
Describe how combinations are used in probability calculations.
In probability, combinations are used to determine the number of ways to select a subset of items from a larger set, where the order of the items does not matter. For example, if you are rolling two six-sided dice, the probability of rolling a specific pair of numbers (e.g., a 3 and a 5) is given by the number of ways to choose 2 numbers from 6 possible numbers, which is $\binom{6}{2}$. Combinations are essential for calculating probabilities of events, as they provide the necessary information about the number of possible outcomes.
Analyze the relationship between combinations and permutations.
Combinations and permutations are related but distinct concepts in mathematics. While permutations involve arranging a set of items in a specific order, combinations involve selecting a subset of items from a larger set, where the order of the selected items does not matter. The formula for the number of combinations of r items from a set of n items, $\binom{n}{r}$, can be expressed in terms of permutations as $\frac{n!}{r!(n-r)!}$, where $n!$ represents the number of permutations of n items. This relationship highlights the fundamental differences between these two important concepts in mathematics.
The binomial coefficient, also known as 'choose' or 'nCr', represents the number of ways to choose r items from a set of n items, where order does not matter.