My-mother-is-making-a-game-for-children-that-involves-words-on-cards-e-g-card-one-hase-words-a-and-b-for-whatever-a-and-b-might-be-She-has-a-list-of-20-words-and-she-wants-cards-that-are-t-duplic

Question Number 5315 by FilupSmith last updated on 07/May/16

My mother is making a game for children

that involves words on cards .

e . g . card one hase words a and b,

for whatever a and b might be .

She has a list of 20 words and she wants cards

that {are}^{'} t duplicate . That is, a b and b a

are duplicates .

My question is, the total number of cards

is equal to^{20} C_{2}, correct ?

Commented by Yozzii last updated on 07/May/16

N o d u p l i c a t i o n m e a n s o r d e r d o e s n o t

m a t t e r . S o, c o m b i n a t i o n s a r e r e q u i r e d .

T w o d i f f e r e n t w o r d s f r o m a l i s t o f

20 w o r d s (a s s u m e d d i f f e r e n t) a r e

g i v e n p e r c a r d . T h u s, \frac{20}{2} = 10 c a r d s (c a r d s b e i n g d e f i n e d b y h a r d p a p e r o n l y) a r e

n e e d e d . T o t h e f i r s t c a r d t h a t y o u u s e, y o u h a v e (\begin{matrix} 20 \\ 2 \end{matrix})

c h o i c e s f o r y i e l d i n g t h e f i r s t c a r d .

2 w o r d s a r e r e m o v e d b y t h i s a c t i o n

s o t h a t t h e s e c o n d c a r d y o u u s e g e t s (\begin{matrix} 18 \\ 2 \end{matrix}) w o r d c h o i c e s .

2 m o r e w o r d s a r e l o s t . T o t h e t h i r d

c a r d y o u u s e \Rightarrow (\begin{matrix} 16 \\ 2 \end{matrix}) c h o i c e s o r (\begin{matrix} 16 \\ 2 \end{matrix})

p o s s i b l e c a r d s r e s u l t . C o n t i n u i n g o n,

y o u e v e n t u a l l y h a v e j u s t 2 w o r d s l e f t

f o r c a r d t e n i n (\begin{matrix} 2 \\ 2 \end{matrix}) = 1 w a y .

B y t h i s c a r d g e n e r a t i o n m e t h o d, a t r a n d o m, t h e r e

a r e e x a c t l y \underset{i = 1}{\prod^{10}} (\begin{matrix} 22 - 2 i \\ 2 \end{matrix}) w a y s .

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