
Words are formed by arranging the letters of the word “STRANGE” in all possible manners. Let be the number of words in which vowels do not come together and be the number of words in which vowels come together. Then find the ratio of . (Write the ratio in terms of coprime natural numbers)
Answer
481.2k+ views
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 . Then the difference between the above two values will be . 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 distinct objects
Where,
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
Let us call it
Thus,
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
Also, the vowels A and E can be arranged in themselves without separating them in 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
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.
Then the ratio can be written as
Hence the required ratio is
So, the correct answer is “ ”.
Note: You can compare the logic used in the above example to find 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.
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
Where,
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
Let us call it
Thus,
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
Also, the vowels A and E can be arranged in themselves without separating them in
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
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.
Then the ratio
Hence the required ratio is
So, the correct answer is “
Note: You can compare the logic used in the above example to find
Latest Vedantu courses for you
Grade 9 | CBSE | SCHOOL | English
Vedantu 9 CBSE Pro Course - (2025-26)
School Full course for CBSE students
₹37,300 per year
Recently Updated Pages
Earth rotates from West to east ATrue BFalse class 6 social science CBSE

The easternmost longitude of India is A 97circ 25E class 6 social science CBSE

Write the given sentence in the passive voice Ann cant class 6 CBSE

Convert 1 foot into meters A030 meter B03048 meter-class-6-maths-CBSE

What is the LCM of 30 and 40 class 6 maths CBSE

What is history A The science that tries to understand class 6 social science CBSE

Trending doubts
Where did Netaji set up the INA headquarters A Yangon class 10 social studies CBSE

A boat goes 24 km upstream and 28 km downstream in class 10 maths CBSE

Why is there a time difference of about 5 hours between class 10 social science CBSE

The British separated Burma Myanmar from India in 1935 class 10 social science CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

What are the public facilities provided by the government? Also explain each facility
