The word â€˜SOCIETYâ€™ contains seven distinct letters and
they can be arranged at random in a row in $^7p_7$ ways, i.e.
in 7! = 5040 ways
Let us now consider those arrangements in which all the
three vowels come together. So in this case we have to
arrange four letters. S,C,T,Y and a pack of three vowels
in a row which can be done in $^5p_5$ i.e. 5! = 120 ways.
Also, the three vowels in their pack can be arranged in $^3p_3$ i.e. 3! = 6 ways.
Hence, the number of arrangements in which the three
vowels come together is 120 Ã— 6 = 720
âˆ´ The probability that the vowels come together = $720\over5040$=$1\over7$