Question

Find the probability in a random arrangement of letters of the word ‘ SOCIAL’ vowels come together.

Answer 1

Hint: First see the total number of letters. Then see how many letters are actually to be arranged, as the vowels are to be placed together, therefore consider them as one letter. Keeping in mind that these three vowels can also be arranged in different ways.

Complete step-by-step answer:
We have,
Three vowels and three other letters, as the vowels are to be together , consider them as one object.
Therefore there are 4 objects totally, which can be arranged in 4! Ways, but the vowels can be arranged internally in 3! Ways.
Therefore the total number of ways of arranging the word with the vowels together is 4! × 3!
=4×3×2×1×3×2×1
= 24 × 6
=144
i.e. no. of words with vowels together is 144.
But the total no. of random arrangements are = 6! =6×5×4×3×2×1 = 720
Required probability =number of favourable outcomes divided by number of all possible outcomes
\[\begin{gathered}
   = \dfrac{{144}}{{720}} \\
   = \dfrac{1}{5} \\
\end{gathered} \]

Note: Permutation (ways of arrangement) of a set is an arrangement of its elements into a sequence or linear order, or if it is already ordered, a rearrangement of it’s elements. In this question also we rearranged the letters of the word ‘SOCIAL’ which is acting as a set.