Find the probability that a student arranging the letters of the word MATHEMATICS will make all the vowels be together in any arrangement he or she does.


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Question Number 56025 by pieroo last updated on 08/Mar/19

$Find the probability that a student$ $arranging the letters of the word$ $MATHEMATICS will make all the$ $vowels be together in any arrangement$ $he or she does .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 08/Mar/19

	$A = 2 C = 1 E = 1 H = 1 I = 1 M = 2 S = 1 T = 2$ $(A A E I) C H M M S T T$ $(\frac{8!}{2! \times 2!}) \times \frac{4!}{2!} \to i t i s t h e a n s w e r$ $\frac{8 \times 7 \times 6 \times 5 \times 3 \times 2}{1} \times \frac{4 \times 3}{1} = 20160 \times 6 = 120960$ $d i s c u s s i o n$ $a s s u m e d v o w e l A A E I a r e F a b i q u i c k e d$ $p e r m u t a t i o n f o r A A E I$ $i s = \frac{4}{2!} d e n o m i n a t o r 2! f o r t w o A$ $p e r m u t a t i o n f o r (A A E I) C H M M S T T$ $\frac{8!}{2! \times 2!} \to d e n o m i n a t o r 2! f o r t w o (02) M$ $a n i t h e r 2! f o r t w o T$ $s o a n s w e r = \frac{8!}{2! 2!} \times \frac{4!}{2!}$ $y e s i f o r g o t t o f i n d p r o b a b l i t y . . . M r W s i r d o n e i t$ $p l s c h e c k . . .$
	Answered by mr W last updated on 08/Mar/19

	$A A E I = 4 l e t t e r s$ $M M T T H C S = 7 l e t t e r s$ $p = \frac{\frac{8!}{2! 2!} \times \frac{4!}{2!}}{\frac{11!}{2! 2! 2!}} = \frac{8! 4!}{11!} = \frac{4}{165} \approx 2 . 4 %$
	Commented by pieroo last updated on 08/Mar/19

	$Thanks sir,$
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