Problem 48 - Algebra and Trigonometry

Question

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

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

Step 1: 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.

Step 2: 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.

Step 3: 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.