Question

How many words, with or without meaning can be formed using all the letters of the word EQUATION, using each letter exactly once?

Answer 1

Hint: In order to solve such questions where we need total number of words or indirectly total number of arrangements, use the concept of permutation. Use the formula for counting the number of words.

Complete step-by-step answer:
Given word is: EQUATION
Number of letters in the given word = 8
We need to use all the letters at a time. So,
Numbers of letters to be used = 8
As we know that number of different possible arrangements of “n” given items taken “m” at a time is given by ${}^n{P_m}$
For the given case:
Number of given items = n = 8
Number of items taken at a time = m = 8
So, total number of words to be formed $ = {}^n{P_m} = {}^8{P_8}$
Now let us solve the term.
Total number of words to be formed

\[  = {}^8{P_8}  \]

\[   = \dfrac{{8!}}{{\left( {8 - 8} \right)!}}\] 

\[  = \dfrac{{8!}}{{0!}}  \]

\[  = \dfrac{{8!}}{1}{\text{ }}\left[ {\because 0! = 1} \right]  \]
\[  = 8!  \]
\[  = 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1  \]
\[  = 40320  \]

Hence, 40320 words with or without meaning can be formed using all the letters of the word EQUATION, using each letter exactly once.

Note: A permutation is a collection or a combination of objects from a set where the order or the arrangement of the chosen objects does matter. In other words, a permutation is an arrangement of objects in a definite order. Students must remember different formulas for finding the permutation. And also must take extra care when in such problems the letters are repeating.