Video: Solving Problems Involving Permutations and Combinations

Given š‘‹ = {š‘„: š‘„ āˆˆ ā„¤, āˆ’17 ā‰¤ š‘„ < 23} and š‘Œ = {(š‘Ž, š‘, š‘): š‘Ž, š‘, š‘ āˆˆ š‘‹, š‘Ž, š‘, š‘ are distinct elements}, determine š‘›(š‘Œ), the number of elements that belong to the set š‘Œ.

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Video Transcript

Given set š‘‹ is the š‘„-values where š‘„ is an integer greater than or equal to negative 17 and less than 23 and set š‘Œ is the values š‘Ž, š‘, and š‘, where š‘Ž, š‘, and š‘ exist in set š‘‹ and š‘Ž, š‘, and š‘ are distinct elements, determine š‘› of š‘Œ, the number of elements that belong to the set š‘Œ.

We are told that set š‘‹ contains the integers greater than or equal to negative 17 and less than 23. This means that š‘‹ contains the integers negative 17, negative 16, negative 15, and so on, all the way up to 22. There are a total of 40 elements in set š‘‹. Set š‘Œ consists of all the possible permutations of three elements from set š‘‹ where order matters. For example, zero, one, two is counted as different from one, two, zero.

One way of calculating the number of elements in set š‘Œ would be using the fundamental counting principle. As there are 40 elements in set š‘‹, the value of š‘Ž could be any one of these 40 elements. As š‘Ž, š‘, and š‘ are distinct, there are then 39 possible values of š‘. We have now selected two of the elements from set š‘‹, so there are 38 possible values of š‘. The total number of elements that belong to the set š‘Œ will be equal to 40 multiplied by 39 multiplied by 38. This is equal to 59,280.

An alternative method here would be to use our knowledge of permutations. As with any permutation, order matters. And in this case, there is no repetition. š‘›Pš‘Ÿ is therefore equal to š‘› factorial divided by š‘› minus š‘Ÿ factorial. There are 40 elements in total, and we are selecting three of them each time. This means that we need to calculate 40 factorial divided by 37 factorial. We can rewrite the numerator as 40 multiplied by 39 multiplied by 38 multiplied by 37 factorial. Dividing through by 37 factorial, we once again get 40 multiplied by 39 multiplied by 38. This confirms the answer of 59,280. This is the number of elements that belong to the set š‘Œ.

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