let-S-a-b-c-d-e-f-if-we-take-any-subset-S-same-subset-is-allowed-it-also-can-be-S-which-will-form-S-if-we-join-them-order-of-operation-does-not-matter-a-b-c-d-d-e-f-is-the-same-as-

Question Number 190602 by uchihayahia last updated on 07/Apr/23

l e t S = {a, b, c, d, e, f}

i f w e t a k e a n y s u b s e t S (s a m e s u b s e t i s a l l o w e d),

i t a l s o c a n b e S, w h i c h w i l l f o r m S i f w e j o i n t h e m,

o r d e r o f o p e r a t i o n d o e s n o t m a t t e r

({a, b, c, d}, {d, e, f}) i s t h e s a m e a s

({d, e, f}, {a, b, c, d})

h o w m a n y w a y s c a n w e c h o o s e ?

Answered by mr W last updated on 07/Apr/23

{\begin{cases} 1 \\ 6 \end{cases} \begin{matrix}  \end{matrix}} + {\begin{cases} 2 \\ 6 \end{cases} \begin{matrix}  \end{matrix}} + {\begin{cases} 3 \\ 6 \end{cases} \begin{matrix}  \end{matrix}} + {\begin{cases} 4 \\ 6 \end{cases} \begin{matrix}  \end{matrix}} + {\begin{cases} 5 \\ 6 \end{cases} \begin{matrix}  \end{matrix}} + {\begin{cases} 6 \\ 6 \end{cases} \begin{matrix}  \end{matrix}}

= 1 + 31 + 90 + 65 + 15 + 1

= 203

(= B_{6} = 203)

w i t h

{\begin{cases} k \\ n \end{cases} \begin{matrix}  \end{matrix}} = S (n, k) = S t i r l i n g n u m b e r s o f 2 . k i n d

B_{n} = B e l l n u m b e r s

Commented by uchihayahia last updated on 07/Apr/23

f o r e x a m p l e

s u b s e t w i t h 0 e l e m e n t

a n d s u b s e t w i t h 6 e l e m e n t s t h e r e i s 1 c h o i c e

s u b s e t w i t h 1 e l e m e n t

a n d s u b s e t w i t h 5 e l e m e n t s t h e r e i s 6 c h o i c e s

s u b s e t w i t h 1 e l e m e n t

a n d s u b s e t w i t h 6 e l e m e n t s t h e r e i s 6 c h o i c e s

s u b s e t w i t h 2 e l e m e n t

a n d s u b s e t w i t h 6 e l e m e n t s t h e r e i s 15 c h o i c e s

s u b s e t w i t h 2 e l e m e n t

a n d s u b s e t w i t h 5 e l e m e n t s t h e r e i s 30 c h o i c e s

s u b s e t w i t h 2 e l e m e n t

a n d s u b s e t w i t h 4 e l e m e n t s t h e r e i s 15 c h o i c e s

s o o n a n d s o f o r t h

Commented by mr W last updated on 07/Apr/23

Commented by uchihayahia last updated on 07/Apr/23

t h e s u b s e t c a n b e e m p t y s e t, a l s o

a n y s u b s e t t a k e n {d o e s n}^{'} t h a v e t o d i s j o i n t .

i t h i n k {i t}^{'} s m o r e t h a n 52 . i^{'} m s o r r y i f

m y q u s e t i o n w a s n o t c l e a r

Commented by mr W last updated on 07/Apr/23

i^{'} m s o r r y t h a t i t i s s t i l l n o t c l e a r t o

m e . c a n y o u e x p l a i n w i t h s o m e

e x a m p l e s w h a t i s n o t i n c l u d e d i n

m y s o l u t i o n a n d w h a t y o u m e a n

w i t h e m p t y s u b s e t s ?

b t w : i m i s r e a d t h a t t h e s e t h a s o n l y

5 e l e m e n t s . b u t i t h a s 6 e l e m e n t s .

t h i s i s f i x e d .

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