The number of ways that we can permute n items into 5 spots is _______ times the number of ways that we can combine n items into 5 spots.
Question
The number of ways that we can permute n items into 5 spots is _______ times the number of ways that we can combine n items into 5 spots.
Solution
The number of ways we can permute n items into 5 spots is given by the formula for permutations, which is nPr = n! / (n - r)!, where n is the total number of items and r is the number of spots. In this case, r = 5, so the formula becomes nP5 = n! / (n - 5)!.
The number of ways we can combine n items into 5 spots is given by the formula for combinations, which is nCr = n! / [r!(n - r)!], where n is the total number of items and r is the number of spots. In this case, r = 5, so the formula becomes nC5 = n! / [5!(n - 5)!].
To find how many times the number of permutations is greater than the number of combinations, we divide the formula for permutations by the formula for combinations:
nP5 / nC5 = [n! / (n - 5)!] / [n! / 5!(n - 5)!]
Simplifying this expression, we find that the number of permutations is 5! = 120 times the number of combinations.
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