WEBVTT
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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 is the values š, š, and š, where š, š, and š exist in set š and š, š, and š are distinct elements, determine š of š, the number of elements that belong to the set š.
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We are told that set š contains the integers greater than or equal to negative 17 and less than 23.
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This means that š contains the integers negative 17, negative 16, negative 15, and so on, all the way up to 22.
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There are a total of 40 elements in set š.
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Set š consists of all the possible permutations of three elements from set š where order matters.
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For example, zero, one, two is counted as different from one, two, zero.
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One way of calculating the number of elements in set š would be using the fundamental counting principle.
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As there are 40 elements in set š, the value of š could be any one of these 40 elements.
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As š, š, and š are distinct, there are then 39 possible values of š.
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We have now selected two of the elements from set š, so there are 38 possible values of š.
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The total number of elements that belong to the set š will be equal to 40 multiplied by 39 multiplied by 38.
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This is equal to 59,280.
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An alternative method here would be to use our knowledge of permutations.
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As with any permutation, order matters.
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And in this case, there is no repetition.
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šPš is therefore equal to š factorial divided by š minus š factorial.
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There are 40 elements in total, and we are selecting three of them each time.
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This means that we need to calculate 40 factorial divided by 37 factorial.
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We can rewrite the numerator as 40 multiplied by 39 multiplied by 38 multiplied by 37 factorial.
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Dividing through by 37 factorial, we once again get 40 multiplied by 39 multiplied by 38.
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This confirms the answer of 59,280.
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This is the number of elements that belong to the set š.