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Chapter 8: Problem 48

Use the formula for \(_{n} P_{r}\) to solve Exercises \(41-48\) How many arrangements can be made using four of the letters of the word COMBINE if no letter is to be used more than once?

Short Answer

Expert verified
Therefore, there are 840 different arrangements that can be made using four of the letters of the word COMBINE, with no letter used more than once.

Step by step solution

01

Identify n and r

You are asked to make arrangements using four letters of the word COMBINE which has 7 unique letters. So, \(n = 7\) (the total number of letters) and \(r = 4\) (the number of letters to arrange).
02

Use the Permutation Formula

The formula to find the number of permutations is \(_{n} P_{r} = \frac{n!}{(n-r)!}\). Plugging \(n = 7\) and \(r = 4\) into the formula, you get \(_{7} P_{4} = \frac{7!}{(7-4)!}\).
03

Evaluate the Factorials and Simplify

First, calculate the factorials. \(7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5040\) and \((7-4)! = 3! = 3 * 2 * 1 = 6\). Then, simplify the formula: \(_{7} P_{4} = \frac{5040}{6} = 840.

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