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Topic: Permutation and Combination

Permutation and combination are mathematical concepts that deal with counting and arranging objects in a specific order. These concepts are widely used in various fields, including mathematics, probability theory, and computer science.
Permutation refers to the arrangement of objects in a specific order. It is often denoted by the symbol "P". The number of permutations of a set of objects is given by the formula nPr = n! / (n - r)!, where n is the total number of objects and r is the number of objects to be arranged.
Combination, on the other hand, refers to the selection of objects without considering the order. It is often denoted by the symbol "C". The number of combinations of a set of objects is given by the formula nCr = n! / (r! * (n - r)!), where n is the total number of objects and r is the number of objects to be selected.
Understanding permutation and combination is crucial in solving problems related to probability, counting principles, and combinatorial optimization. These concepts are often used in various real-life scenarios, such as arranging a group of people in a line, selecting a committee from a larger group, or finding the number of possible outcomes in a game.

Example

Let's consider an example to better understand permutation and combination. Suppose we have a group of 5 people and we want to arrange them in a line. The number of permutations of arranging these 5 people in a line can be calculated using the permutation formula as follows:
nPr = 5! / (5 - 1)! = 5! / 4! = 5 * 4 * 3 * 2 * 1 / 4 * 3 * 2 * 1 = 5
So, there are 5 different ways to arrange these 5 people in a line.

Exercise

  1. A committee needs to be formed from a group of 10 people. How many different combinations of the committee can be formed if the committee consists of 3 people?
  1. In a race, there are 8 participants. How many different ways can the top 3 positions be filled?

Resources

 
AP CSA: inheritance in JavaA Level 9709:Binomial Expansion
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