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Chapter 5: Problem 56

How many different nine-letter words (real or imaginary) can be formed from the letters in the word ECONOMICS?

Short Answer

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90720

Step by step solution

01

Identify the Letters in ECONOMICS

The word 'ECONOMICS' consists of the letters: E, C, O, N, O, M, I, C, S. Notice that the letter 'C' appears twice and the letter 'O' appears twice.
02

Calculate the Total Permutations

If all the letters were unique, the number of permutations would be calculated by the factorial of the length of the word: \[ 9! = 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 = 362880 \]
03

Account for Repeated Letters

Since 'C' and 'O' appear twice, adjust the total permutations by dividing by the factorial of the number of occurrences of each repeated letter. This adjustment ensures we do not overcount permutations that are actually identical. The formula is: \[ \frac{9!}{2! \times 2!} = \frac{362880}{2 \times 2} = \frac{362880}{4} = 90720 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factorials
A factorial, represented by an exclamation mark after a number, is a product of all positive integers up to that number. For example, we write \( 9! \) to mean \( 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 \).
This concept is crucial when calculating permutations. It tells us how many possible ways we can arrange a set of items.
For instance, if we have 3 letters, say A, B, and C, the number of ways to arrange them is \( 3! = 6 \).
  • ABC
  • ACB
  • BAC
  • BCA
  • CAB
  • CBA

    • In our example with the word 'ECONOMICS', without considering repeated letters, we calculate \( 9! = 362880 \).
    This high number represents the total arrangements of all nine letters.
    Combinatorics
    Combinatorics is the branch of mathematics concerning the counting, arrangement, and combination of objects.
    Applied to the word 'ECONOMICS', combinatorics helps us figure out the total number of unique arrangements for the letters.
    At first glance, if all letters were unique, we use factorials to calculate permutations. However, real-life scenarios often involve repeated elements.
    This is where combinatorial formulas adjust calculations for duplicates ensuring accurate counts.
    For letters in 'ECONOMICS', we initially find permutations: \( 9! = 362880 \), then adjust for two 'C's and two 'O's using: \( \frac{9!}{2! \times 2!} \). This adjustment accounts for repetitions.
    Repeated Letters
    Repeated letters in permutations impact the total count of unique arrangements. When letters repeat, it reduces the number of distinct permutations because swapping identical letters doesn't create new combinations.
    To account for this, we modify the total permutations using a specific formula.
    First, calculate the factorial of the total letters, like \( 9! \).
    Next, divide by the factorials of the counts of each set of repeated letters. For 'ECONOMICS' with 2 'C's and 2 'O's, this is \( \frac{9!}{2! \times 2!} \). This process ensures we don't overcount arrangements.

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