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Chapter 11: Problem 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
840 different 4-letter arrangements can be made from the word 'COMBINE' without repeating any letter.

Step by step solution

01

Identify Unique Characters

First and foremost, remember that we can't use the same letter twice. This means we only consider unique characters. The word 'COMBINE' has 7 unique characters. This is what we'll be choosing from.
02

Permutation Without Repetition

Next, apply the equation of permutation without repetition, which basically says how many ways we can arrange r elements in a set of n unique characters. The formula for permutation without repetition is:\[ P(n, r) = n! / (n - r)! \]where n is the total number of characters (7 in our case, as the word combine has seven distinct letters), r is the number of characters in one arrangement (4 in this case, as we want to arrange 4 letters), and ! denotes a factorial, a function that multiplies all positive integers less than or equal to the chosen number.
03

Calculate The Permutations

Plug in the values into the formula:\[ P(7, 4) = 7! / (7 - 4)! \]\[ P(7, 4) = 7! / 3! \]\[ P(7, 4) = (7*6*5*4*3*2*1) / (3*2*1) \]\[ P(7, 4) = 7*6*5*4 = 840 \]This gives the total number of 4-letter arrangements we can make from the word 'COMBINE' without repeating letters.

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