Problem 40 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the number of distinguishable permutations that can be formed from the letters of the word PHILIPPINES.

Step-by-Step Solution

Answer

There are 34,650 distinguishable permutations that can be formed from the word PHILIPPINES. This is calculated by dividing the total number of permutations (11!) by the product of the factorials of each repeated letter: \(\frac{11!}{3! \times 3! \times 2!}\).

Step 1: Calculate the total number of permutations

First, we need to find the total number of permutations for an 11-letter word. In general, the number of permutations of a word with n distinct letters is given by n! (factorial). In this case, n=11, so there are 11! permutations.

Step 2: Calculate the number of ways each repeated letter can be arranged

Next, we need to find the factorials of the counts of each of the repeated letters: - Factorial of letter P (3! - since there are 3 occurrences of P) - Factorial of letter I (3! - since there are 3 occurrences of I) - Factorial of letter L (2! - since there are 2 occurrences of L)

Step 3: Calculate the number of distinguishable permutations

Now, we will divide the total number of permutations (11! - calculated in step 1) by the product of the factorials of each repeated letter (calculated in step 2). Number of Distinguishable Permutations = \(\frac{11!}{3! \times 3! \times 2!}\) Calculate the values: Number of Distinguishable Permutations = \(\frac{11!}{(3! \times 3! \times 2!)} = \frac{11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2}{(6 \times 6 \times 2)}\) Number of Distinguishable Permutations = 34,650 Thus, there are 34,650 distinguishable permutations that can be formed from the word PHILIPPINES.

Key Concepts

Factorials

Understanding factorials is essential for tackling problems related to permutations and combinations. A factorial, denoted by an exclamation mark (e.g., \(!5!\)), is the product of an integer and all the integers below it down to one.
For example, when we say \(!5!\), we mean \(5 \times 4 \times 3 \times 2 \times 1\). In general, for any positive integer \(n\), the factorial is expressed as \(n! = n \times (n-1) \times \ldots \times 2 \times 1\).
Factorials grow very quickly. For instance, \(!11!\) is already 39,916,800, illustrating how exponential the growth is.
Factorials are crucial in permutations, especially when dealing with repeated items. For example, in the word "PHILIPPINES," repeated letters mean we need to adjust our calculations to account for indistinguishable arrangements. We divide the total permutations by the product of factorials of the repeated letters' counts.

Combinatorics

Combinatorics is the branch of mathematics dealing with combinations of objects. It's about determining how many different ways there are to select or arrange these objects.
Permutation and combination questions fall under this category. In permutations, order matters. This is crucial when letters or numbers are being arranged to analyze how many distinct sequences can be formed.
The problem of determining the distinguishable permutations in a word falls into this category. The word "PHILIPPINES" uses 11 letters, but with repetitions of 'P', 'I', and 'L'. So, it involves calculating permutations with repetitions, a classic combinatorics challenge.

• The total permutations of all 11 letters are first calculated as \(11!\).
• The permutations are then adjusted to consider repeated letters by dividing by \(3 !\) for each 'P' and 'I', and \(2!\) for 'L'.

This final calculation gives the number of unique sequences possible from the given set of letters.

Counting Methods

Counting methods are techniques used to evaluate how many ways certain events can occur. These methods are invaluable in probability, algebra, and everyday problems.
In mathematics, problems often require finding the number of different ways to arrange objects or choose subsets.
There are two primary methods: permutations and combinations.
Permutations, as demonstrated in the "PHILIPPINES" problem, are used when arrangement order is important.

• The basic principle of permutation is calculated using factorials to arrange distinct items.
• Incorporating repetition calls for dividing by factorials of the repeated item's occurrences.

Combinations, on the other hand, do not consider order. Counting methods often start by determining if order matters, guiding you to choose the correct approach.