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Question
Find the probability that in a random arrangement of the letters of the word 'SOCIAL' vowels come together.
Solution
There are six letters in the word ‘SOCIAL’, which can be arranged in 6! ways.
There are three vowels, namely O, I and A.
Let us consider these three vowels as one letter.
So, when the three vowels are clubbed together, we have (O, I, A) SCL. We can arrange four letters in a row in 4! ways.
Also, the three vowels can themselves be arranged in 3! ways.
Hence, required probability = \[\frac{4! \times 3!}{6!} = \frac{4! \times 3 \times 2}{6 \times 5 \times 4!} = \frac{1}{5}\]
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