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# If the letters of the word 'ASSASSINATION' are arranged at random in a row, find the probability that (a) four S's come consecutively in the word (b) All A's not coming together (c) two I's and two N's come together.

$(A)\; \frac{2}{143} , \frac{1}{26} , \frac{2}{143}$
$(B)\; \frac{1}{143} , \frac{2}{26} , \frac{2}{143}$
$(C) \; \frac{3}{143}, \frac{1}{26}, \frac{2}{143}$
$(D)\; \frac{1}{143}, \frac{1}{26}, \frac{1}{143}$

The number of arrangements in which all the letters of the given words can be arranged = $\frac{13 !}{4 ! 3 !2! 2 !}$
(a) When all four S's come together, then the no. of arrangement = $\frac{10 !}{3 ! 2 ! 2 !}$
$\therefore$ Required probability = $\frac{\frac{10!}{3 ! \;2!\;2!}}{\frac{13!}{4!\; 3!\; 2!\; 2!}} = \frac{10! 4!}{13!}$
$\; \; \; \; \; \; \; \; = \frac{2}{143}$
(b) The no. of arrangements in which all A's come together = $\frac{11!}{ 4! 2! 2!}$
$\therefore$ Probability of all A's coming together = $\frac{\frac{11!}{4! \; 2! \; 2!}}{\frac{13!}{4! \; 3! \; 2!\; 2!}} = \frac{11! \times 3!}{13!}$
$\; \; \; \; \; \; \; \; \; \; = \frac{1}{26}$
(c) The no. of arrangement in which two I's and two N's come together = $\frac{10!}{4! \; 3!} \times \frac{4!}{2! \times 2!} = \frac{10!}{3! \;2! \; 2!}$
$\therefore$ Required probability = $\frac{\frac{10!}{3! \; 2! \; 2!}}{ \frac{13!}{4! \; 3! \ 2! \;2! }} = \frac{10! \times 4 !}{13!}$
$\; \; \; \; \; \; \; \; \; \; \; = \frac{2}{124}$