In a random arranngement of the letters of the word " COMMERCE " , find the probability that all vowels - Maths - Probability

Question

In a random arranngement of the letters of the word " COMMERCE "
, find the probability that all vowels come together

Shridhar Kaushik · Accepted Answer

In the given world COMMERCE, repeating letters are C, M, E(each is repeating twice)

$S o, Total no. of exhaustive cases = \frac{8!}{2! \times 2! \times 2!} = \frac{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1 \times 2 \times 1} = 5040$

Let A be the event that the vowels come together.

So the word will be of the form C(OEE)MMRC

The number of favourable cases = {Number of ways in which 6 letters (taking vowels as a unit) can be arranged} × (Number of ways in which vowels can be interchanged between themselves)

$= \frac{6!}{2! \times 2!} \times \frac{3!}{2!} = \frac{6 \times 5 \times 4 \times 3 \times 2 \times 1}{2 \times 1 \times 2 \times 1} \times \frac{3 \times 2 \times 1}{2 \times 1} = 540$

Hence, the probability of getting vowels together is–

$P (A) = \frac{No. of favourable cases}{Total No. of exhaustive cases} = \frac{540}{5040} = \frac{3}{28}$

In a random arranngement of the letters of the word " COMMERCE " , find the probability that all vowels come together ..