Lecture08-12Sept08

# Lecture08-12Sept08 - P r o p e r t ie s o f n C r ) ! (! !...

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Unformatted text preview: P r o p e r t ie s o f n C r ) ! (! ! r n r n r n C r n = = = = + n r r n r n y x r n y x ) ( T h is is c a l le d t h e b in o m ia l e x p a n s io n a n d n C r a r e c a l le d t h e b in o m ia l c o e f f ic ie n t s . n- 1 C r + n- 1 C r- 1 = n C r P r o o f : P r o o f : M o r e a b o u t n C r n C r = n u m b e r o f w a y s o f c h o o s in g r o b je c t s f r o m a s e t o f n w i th o u t r e p la c e m e n t i f o r d e r d o e s n o t m a t te r S e p t . 2 8 A n o t h e r L o o k a t n C r S u p p o s e w e h a v e n o b je c t s , r o f o n e t y p e a n d n- r o f t h e o t h e r . H o w m a n y w a y s c a n w e a r r a n g e t h e m in a l in e ? A n o t h e r L o o k a t n C r S u p p o s e w e h a v e n o b je c t s , r o f o n e t y p e a n d n- r o f t h e o t h e r . H o w m a n y w a y s c a n w e a r r a n g e t h e m in a l in e ? id e a 1 : g iv e a l l t h e o b je c t s u n iq u e la b e ls , a r r a n g e t h e m a n d t h e n s e e h o w m a n y a r r a n g e m e n t s a r e t h e s a m e b y p e rm u t in g t h e id e n t ic a l o b je c t s A 1 A 2 B 1 A 3 B 2 is t h e s a m e a s A 3 A 2 B 2 A 1 B 1 A n o t h e r L o o k a t n C r S u p p o s e w e h a v e n o b je c t s , r o f o n e t y p e a n d n- r o f t h e o t h e r . H o w m a n y w a y s c a n w e a r r a n g e t h e m in a l in e ? id e a 1 : g iv e a l l t h e o b je c t s u n iq u e la b e ls , a r r a n g e t h e m a n d t h e n s e e h o w m a n y a r r a n g e m e n t s a r e t h e s a m e...
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## This note was uploaded on 09/16/2008 for the course STAT 414 taught by Professor Senturk,damla during the Fall '07 term at Pennsylvania State University, University Park.

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Lecture08-12Sept08 - P r o p e r t ie s o f n C r ) ! (! !...

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