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 co prime natural numbers)

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\frac{5}{2}

\frac{7}{9}

\frac{6}{5}

\frac{8}{4}

answer is A.

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Detailed Solution

The given word : “STRANGE”
There are a total of 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!
=7×6×5×4×3×2×1
=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!
=6×5×4×3×2×1
=720
Also, the vowels A and E can be arranged in themselves without separating them in 2!=2×1=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=2×720
⇒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
⇒m=5040−1440
⇒m=3600
Then the ratio m:n can be written as