100 students are standing in a row. 4 students should be selected in the way that no two of them are next to each other. how many ways do you have to do this?


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Question Number 103406 by mr W last updated on 16/Jul/20

$100 s t u d e n t s a r e s t a n d i n g i n a r o w .$ $4 s t u d e n t s s h o u l d b e s e l e c t e d i n t h e$ $w a y t h a t n o t w o o f t h e m a r e n e x t t o$ $e a c h o t h e r . h o w m a n y w a y s d o y o u$ $h a v e t o d o t h i s ?$
Commented by bobhans last updated on 15/Jul/20

$n (S) = C_{4}^{100} = \frac{100!}{4! . 96!}$ $n (A) = 100! - 50 . C_{2}^{50} = 100! - 50 \times \frac{50!}{2! . 48!}$ $p (A) = \frac{100! - 50 . \frac{50!}{2.48!}}{100!} \times 4! . 96!$ $= \frac{100! - \frac{50.50!}{2.48!}}{100.99.98.97} \times 4! = \frac{100! - \frac{50.50.49}{2}}{100.99.98.97} \times 4!$
Commented by mr W last updated on 16/Jul/20

$3 464 840 w a y s$
Commented by mr W last updated on 16/Jul/20

$a n s w e r s e e Q 103716$
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