Question

Words are formed by arranging the letters of the word “STRANGE” in all possible manners. Let $ m $ be the number of words in which vowels do not come together and $ n $ be the number of words in which vowels come together. Then find the ratio of $ m:n $ . (Write the ratio in terms of coprime natural numbers)

Answer 1

Hint: Use the formula of permutation to find the total number of possible ways of arranging the letters of the word “STRANGE”. Then consider the vowels as one unit and again use the same formula to find $ n $ . Then the difference between the above two values will be $ m $ . Then find the required ratio.

Complete step-by-step answer:
The given word in the question is “STRANGE”
There are total 7 letters in the word.
Now, we have the formula,
Total number of arrangement of $ n $ distinct objects $ = n! $
Where, $ n! = n(n - 1)(n - 2)(n - 3)....3.2.1 $
Since, all the letters in the word “STRANGE” are different, we can use the above formula and write,
Total number of arrangement of the letters of the word “STRANGE” in all possible manner $ = 7! $
 $ = 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1 $
 $ = 5040 $
Let us call it $ t $
Thus, $ t = 5040 $
Now, we have to find the number of words in which vowels are together.
Vowels in the word “STRANGE” are: A, E
So, there are two vowels.
To find the total number of words in which vowels are together, we will combine the given vowels as one unit.
So, after taking, AE as one unit. The number of units we have are 6.
Hence, the possibilities of arranging them $ = 6! $
 $ = 6 \times 5 \times 4 \times 3 \times 2 \times 1 $
 $ = 720 $
Also, the vowels A and E can be arranged in themselves without separating them in $ 2! = 2 \times 1 = 2 $ ways.
Thus the total number of possibilities of writing the letters of the word “STRANGE” in such a way that the vowels are always together is given by
 $ n = 2 \times 720 $
 $ \Rightarrow n = 1440 $
Now, if we observe closely, we can say that the total number of possibilities of writing the letters of the word “STRANGE” in such a way that the vowels are not together is the difference between the total number of possible ways of arranging the letters of the word “STRANGE” and the total number of possible ways of arranging the letters of the word “STRANGE” when the vowels are always together. i.e.
 $ m = t - n $
 $ \Rightarrow m = 5040 - 1440 $
 $ \Rightarrow m = 3600 $
Then the ratio $ m:n $ can be written as
 $ \dfrac{m}{n} = \dfrac{{3600}}{{1440}} = \dfrac{5}{2} $
Hence the required ratio is $ \dfrac{5}{2} $
So, the correct answer is “ $ \dfrac{5}{2} $ ”.

Note: You can compare the logic used in the above example to find $ m $ with the logic we use in finding the area of a shaded region after cutting some area from the total area. In those examples, we always find the total area and then subtract the area that we cut from the total area to find the area of the shaded region. Similarly, in this question, we calculated the total number of possible arrangements, then subtracted the number of possible arrangements of taking vowels together, to find the possible arrangements of ‘not having’ the vowels together.