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Explain the concept of combination in mathematics.
A combination in mathematics is a selection of items from a set with distinct members, where the order of selection does not matter. It is a subset of k distinct elements of the set S, and two combinations are identical if and only if they have the same members.
Define the number of k-combinations and its notation.
The number of k-combinations, denoted by $C(n, k)$ or $C_{k}^{n}$, is the binomial coefficient representing the number of ways to choose k elements from a set of n elements. It can be written using factorials as $C(n, k) = \frac{n!},{k!(n-k)!}$.
What is the condition for the binomial coefficient C(n, k) to be defined?
The binomial coefficient $C(n, k)$ is defined whenever $k \leq n$, and it is zero when $k > n$.
How can the number of k-combinations be calculated using factorials?
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Provide an example of a real-life scenario where combinations are used.
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Study Notes
Combinations in Mathematics
- A combination is a selection of items from a larger collection, where the order of the items does not matter.
- Combinations are used to count the number of ways to choose a subset of objects from a larger set.
Number of k-Combinations and Notation
- The number of k-combinations is the number of ways to choose k items from a set of n items, denoted as C(n, k) or "n choose k".
- The notation C(n, k) represents the number of combinations of n items taken k at a time.
Condition for Binomial Coefficient C(n, k) to be Defined
- The binomial coefficient C(n, k) is defined only if n is a positive integer and k is an integer such that 0 ≤ k ≤ n.
Calculating the Number of k-Combinations using Factorials
- The number of k-combinations can be calculated using the formula: C(n, k) = n! / (k!(n-k)!)
- This formula is derived from the fact that the number of ways to arrange k items out of n is equal to the total number of permutations of n items divided by the number of ways to arrange k items and the number of ways to arrange the remaining n-k items.
Real-Life Scenario: Combinations in Action
- Example: A committee of 5 people is to be formed from a group of 12 people. How many different committees can be formed?
- In this scenario, the number of combinations (C(12, 5)) is used to calculate the number of possible committees, which is equal to 792.
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Description
Test your understanding of combinations in mathematics with this quiz. Explore the concept of selecting items from a set without considering the order of selection. Practice identifying different combinations and strengthen your mathematical skills.
