Problem 51 - Algebra and Trigonometry

Question

How many different 9-letter words (meaningful or not) can be formed from the letters in the word ECONOMICS?

Step-by-Step Solution

Answer

90720

Step 1: Identify the letters

List all the letters in the word ‘ECONOMICS’. They are E, C, O, N, O, M, I, C, S.

Step 2: Count the occurrences

Note the frequency of each letter: E (1), C (2), O (2), N (1), M (1), I (1), and S (1).

Step 3: Apply the permutation formula

Use the formula for permutations of multiset: \[\frac{n!}{n1! \times n2! \times \text{...} \times nk!}\] where n is the total number of letters and each ni is the frequency of each distinct letter.

Step 4: Calculate the total permutations

Substitute the values: \[ n = 9, n1 = 1! (E), n2 = 2! (C), n3 = 2! (O), n4 = 1! (N), n5 = 1! (M), n6 = 1! (I), n7 = 1! (S)\] so from the given formula we get: \[\frac{9!}{1! \times 2! \times 2! \times 1! \times 1! \times 1! \times 1!} = \frac{362880}{4} = 90720\]

Key Concepts

Permutations

Permutations are arrangements of objects in a specific order. When we talk about permutations in algebra, we mean the different ways we can arrange a set of items. For example, if we have three letters: A, B, C, the possible permutations are ABC, ACB, BAC, BCA, CAB, and CBA. Each permutation is a unique arrangement of the letters. Permutations are essential because they help calculate the number of possible outcomes in many mathematical problems and real-life scenarios.

Factorials

A factorial, denoted by an exclamation point (!), is the product of an integer and all the positive integers below it. For instance, the factorial of 4 (written as 4!) is calculated as 4 × 3 × 2 × 1 = 24. Factorials are a key concept in permutations and combinations because they represent the total number of ways to arrange a set of items. For example, to find the number of ways to arrange 9 letters, we calculate 9!, which equals 362880. Factorials grow very quickly, which makes them useful in various counting problems in mathematics.

Word Formation

Word formation in algebra involves figuring out how many different ways we can arrange letters to form words. These words do not need to be meaningful. In the problem given, we are forming a 9-letter word using the letters from 'ECONOMICS'. By analyzing each letter and its frequency, we can determine the number of possible arrangements using the permutation formula. It's important to note the frequency of repeated letters because it affects the total number of unique permutations.

Multiset Permutations

Multiset permutations take into account the repetition of items within a set. When we have duplicates, the formula to calculate permutations is adjusted. The formula is \ \frac{n!}{n1! \times n2! \times \text{...} \times nk!} \, where \( n \) is the total number of items, and \( n1, n2, \text{...}, nk \) are the frequencies of each distinct item. This formula divides the total number of permutations by the factorials of the frequencies of each repeated item. In the word 'ECONOMICS', C and O are repeated twice each. So, we divide 9! by the factorials of these frequencies: \ \frac{9!}{1! \times 2! \times 2! \times 1! \times 1! \times 1! \times 1!} = \frac{362880}{4} = 90720 \, giving the total number of unique 9-letter arrangements.