If the letters of the word ASSASSINTION are arranged at random. Find the probability that Four Ss come consecutively in the word. Two Is and two Ns come together. All As are not coming together. No two As are coming together.

Explanation

Total number of letters in the word 'ASSASSINATION' are 13.
Out of which `3A's, 4S's, 2I's, 2N's, 1 T's and 10.`
(i) If four S's come consecutively in the word, then we considers these 4 S's as 1 group. Now, the number of laters is 10.
(i) If four S's come consecutively in the word, then we considers these 4 S's as 1 group. Now, the number of laters is 10.

`"Number of words when all S's are together "=(10!)/(3!2!2!)`
`"Total number of word using letters of the word 'ASSASSINATION' "=(13!)/(3!4!2!2!)`
`therefore" ""Required probability "=(10!)/((3!2!2!xx13!)/(3!4!2!2!))`
`(10!xx4!)/(13!)=(4!)/(13xx12xx11)=(24)/(1716)=(2)/(143)`
(ii) If 2 I's and 2 N's come together, then there as 10 alphabets.
Number of word when 2 I's and 2 N's are come together
`(10!)/(3!4!)xx(4!)/(2!2!)`
`therefore " Required probalitiy "=(10!4!)/((3!4!2!2!)/((13!)/(3!4!2!2!)))=(4!10!)/(2!2!3!4!)xx(3!4!2!2!)/(13!)`
`=(4!10!)/(13!)=(4!)/(13xx12xx11)=(24)/(13xx12xx11)=(2)/(143)`
(iii) If all A's are coming together, then there are 11 alphabets.
Number of words when all A's come together `=(11!)/(4!2!2!)`
`"Probability when all A's come together "=(11!)/((4!2!2!)/((13!)/(4!3!2!2!)))=(11!)/(4!2!2!)xx(4!3!2!2!)/(13!)=(11!xx3!)/(13!)=(6)/(13xx13)=(1)/(26)`
`"Required probability when all A's does not come together "=1-(1)/(26)=(25)/(26)`
(iv) If no two A's are together, then first we arrange the alphabets except A's

`"All the alphabets except A's are arranged in " (10!)/(4!2!2!)`
`"There are 11 vacant places between these alphabets."`
`"So, 3 A's can be place in 11 places in ".^(11)C_(3)" ways"=(11!)/(3!8!)`
`therefore "Total number of words when no two A's together "=(11!)/(3!8!)xx(10!)/(4!2!2!)`
`therefore "Required probability "=(11!xx10!)/(3!8!4!2!2!)xx(4!3!2!2!)/(13!)=(10!)/(8!xx13xx12)`
`=(10xx9)/(13xx12)=(90)/(156)=(15)/(26)`