Problem 48 - College Algebra

Question

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?

Step-by-Step Solution

Answer

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 1: 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).

Step 2: 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 \(_{7} P_{4} = \frac{7!}{(7-4)!}\).

Step 3: 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.