MathsPermutations and Combinations – Definition, Difference and Formula

Permutations and Combinations – Definition, Difference and Formula

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Permutations and combinations are mathematical concepts that are used to calculate the number of ways that a set of objects can be arranged or selected. These concepts are used in a variety of fields, including mathematics, computer science, and statistics.

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    Permutations:

    A permutation is an arrangement of a set of objects in a specific order. The number of permutations of a set of n objects is calculated using the formula:

    n!/(n1! * n2! * … * nk!)

    where n! is the factorial of n (n! = n * (n-1) * (n-2) * … * 2 * 1), and n1, n2, …, nk are the number of times that each object appears in the set.

    For example, consider a set of three objects: {A, B, C}. The number of permutations of this set is 3!/(1! * 1! * 1!) = 3!/1! = 3!/1 = 3. The three permutations are:

    {A, B, C} {A, C, B} {B, A, C} {B, C, A} {C, A, B} {C, B, A}

    Combinations:

    A combination is a selection of a set of objects, without regard to the order in which they are selected. The number of combinations of a set of n objects taken k at a time is calculated using the formula:

    n!/(k! * (n-k)!)

    For example, consider a set of three objects: {A, B, C}. The number of combinations of this set taken 2 at a time is 3!/(2! * (3-2)!) = 3!/2! = 3. The three combinations are:

    {A, B} {A, C} {B, C}

     

    Definition of Permutation and Combination

    A permutation is an arrangement of a set of objects in a specific order. For example, the set {A, B, C} has six permutations: {A, B, C}, {A, C, B}, {B, A, C}, {B, C, A}, {C, A, B}, and {C, B, A}.

    A combination is a selection of a set of objects, without regard to the order in which they are selected. For example, the set {A, B, C} has three combinations when taken two at a time: {A, B}, {A, C}, and {B, C}.

    Permutations and combinations are used to calculate the number of ways that a set of objects can be arranged or selected. These concepts are useful in a variety of fields, including mathematics, computer science, and statistics.

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    The Difference between Permutations and Combinations

    The main difference between permutation and combination is the order in which the objects are arranged or selected.

    Permutation:

    In a permutation, the order of the objects is important. For example, the set {A, B, C} has six permutations: {A, B, C}, {A, C, B}, {B, A, C}, {B, C, A}, {C, A, B}, and {C, B, A}.

    Combination:

    In a combination, the order of the objects is not important. For example, the set {A, B, C} has three combinations when taken two at a time: {A, B}, {A, C}, and {B, C}. These three combinations are considered to be different from each other, even though they contain the same objects.

    Permutations and combinations are used to calculate the number of ways that a set of objects can be arranged or selected. These concepts are useful in a variety of fields, including mathematics, computer science, and statistics.

     

    Basic Formula :

    Permutation:

    The basic formula for calculating the number of permutations of a set of n objects is:

    n!/(n1! * n2! * … * nk!)

    where n! is the factorial of n (n! = n * (n-1) * (n-2) * … * 2 * 1), and n1, n2, …, nk are the number of times that each object appears in the set.

    For example, consider a set of three objects: {A, B, C}. The number of permutations of this set is 3!/(1! * 1! * 1!) = 3!/1! = 3!/1 = 3. The three permutations are:

    {A, B, C} {A, C, B} {B, A, C} {B, C, A} {C, A, B} {C, B, A}

    Combination:

    The basic formula for calculating the number of combinations of a set of n objects taken k at a time is:

    n!/(k! * (n-k)!)

    For example, consider a set of three objects: {A, B, C}. The number of combinations of this set taken 2 at a time is 3!/(2! * (3-2)!) = 3!/2! = 3. The three combinations are:

    {A, B} {A, C} {B, C}

     

    Permutation and Combination Word Problems

    Here are five permutation and combination word problems:

    1. You have a set of three books: {A, B, C}. In how many ways can you arrange the books on a shelf?

    Solution: This is a permutation problem because the order in which the books are arranged is important. The number of permutations of a set of three objects is 3!/(1! * 1! * 1!) = 3!/1! = 3!/1 = 3. The three permutations are: {A, B, C}, {A, C, B}, and {B, A, C}.

    1. You have a set of four marbles: {R, G, B, W}. In how many ways can you draw three marbles from the set, without replacement?

    Solution: This is a combination problem because the order in which the marbles are drawn is not important. The number of combinations of a set of four objects taken three at a time is 4!/(3! * (4-3)!) = 4!/3! = 4. The four combinations are: {R, G, B}, {R, G, W}, {R, B, W}, and {G, B, W}.

    1. You have a set of five cards: {A, 2, 3, 4, 5}. In how many ways can you draw two cards from the set, without replacement?

    Solution: This is a combination problem because the order in which the cards are drawn is not important. The number of combinations of a set of five objects taken two at a time is 5!/(2! * (5-2)!) = 5!/2! = 10. The ten combinations are: {A, 2}, {A, 3}, {A, 4}, {A, 5}, {2, 3}, {2, 4}, {2, 5}, {3, 4}, {3, 5}, and {4, 5}.

    1. You have a set of six dice: {1, 2, 3, 4, 5, 6}. In how many ways can you roll the dice and get a sum of 9?

    Solution: This is a permutation problem because the order in which the dice are rolled is important. The possible ways to roll the dice to get a sum of 9 are: (6, 3), (5, 4), and (4, 5). There are 3 permutations in total.

     

     

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