11th Com Maths Part 2 Chapter 6 (Digest) Maharashtra state board

Chapter 6 Permutations and Combinations

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Project on Permutations and Combinations

Permutations and combinations are fundamental concepts in mathematics, particularly in the field of combinatorics, which deals with counting and arranging objects.

1.         Permutations:

•             A permutation refers to an arrangement of objects in a specific order.

•             In permutations, the order matters. That means rearranging the elements leads to a different permutation.

•             The number of permutations of a set of 𝑛n distinct objects taken 𝑟r at a time is denoted by 𝑃(𝑛,𝑟)P(n,r) or 𝑛𝑃𝑟nPr, and it is calculated as:

𝑃(𝑛,𝑟)=𝑛!(𝑛−𝑟)!P(n,r)=(n−r)!n!

•             Here, 𝑛!n! represents the factorial of 𝑛n, which is the product of all positive integers up to 𝑛n.

2.         Combinations:

•             A combination is a selection of objects from a set, where the order doesn't matter.

•             In combinations, rearranging the elements doesn't result in a different combination.

•             The number of combinations of a set of 𝑛n distinct objects taken 𝑟r at a time is denoted by 𝐶(𝑛,𝑟)C(n,r) or 𝑛𝐶𝑟nCr, and it is calculated as:

𝐶(𝑛,𝑟)=𝑛!𝑟!⋅(𝑛−𝑟)!C(n,r)=r!⋅(n−r)!n!

•             Here, 𝑛!n! represents the factorial of 𝑛n, 𝑟!r! represents the factorial of 𝑟r, and (𝑛−𝑟)!(n−r)! represents the factorial of 𝑛−𝑟n−r.

In summary, permutations deal with arrangements where order matters, while combinations deal with selections where order doesn't matter. Both permutations and combinations are important in various areas of mathematics, including probability, statistics, and algebra.