How many words can be formed by the letters of the word 'PROPERTIES' taken all at a time?

Understand the Problem

The question is asking for the total number of distinct words that can be formed using all the letters of the word 'PROPERTIES' at once. To solve this, we need to consider the arrangement of letters and account for any repetitions.

Answer

The total number of distinct arrangements is 19,958,400.

Steps to Solve

Count the total letters in "PROPERTIES"

The word "PROPERTIES" has 11 letters.

Identify and count the repeating letters

In "PROPERTIES", the letter 'P' appears 2 times, the letter 'R' appears 1 time, the letter 'O' appears 1 time, the letter 'T' appears 1 time, the letter 'I' appears 1 time, the letter 'E' appears 1 time, and the letter 'S' appears 1 time.

So, the counts are:

P: 2
R: 1
O: 1
T: 1
I: 1
E: 1
S: 1

Use the formula for permutations of multiset

The formula to find the number of distinct arrangements (permutations) of a set of objects where some objects are identical is given by:

$$ \text{Permutations} = \frac{n!}{n_1! \cdot n_2! \cdot \ldots \cdot n_k!} $$

Where:

( n ) = total number of letters = 11
( n_1, n_2, \ldots, n_k ) = counts of each repeating letter

For "PROPERTIES":

$$ \text{Permutations} = \frac{11!}{2! \cdot 1! \cdot 1! \cdot 1! \cdot 1! \cdot 1! \cdot 1!} $$

Calculate the factorial values

Calculate ( 11! ) and ( 2! ):

$$ 11! = 39916800 $$ $$ 2! = 2 $$

Calculate the total permutations

Substituting these values in the formula:

$$ \text{Permutations} = \frac{39916800}{2} = 19958400 $$

The total number of distinct words that can be formed from the letters of the word "PROPERTIES" is 19,958,400.

More Information

The calculation considers all possible arrangements of the letters, accounting for the repetition of the letter 'P'. Such problems are common in combinatorics and highlight the importance of understanding permutations in scenarios involving identical items.

Tips

Forgetting to account for repeating letters properly can lead to incorrect calculations. Always ensure to include the correct counts in the factorial denominators.
Miscalculating the factorial of large numbers might occur; ensure the calculations are performed carefully.

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