How-many-sets-of-3-numbers-each-can-be-formed-from-the-numbers-1-2-3-20-if-no-two-consecutive-numbers-are-to-be-in-a-set-

Question Number 118694 by benjo_mathlover last updated on 19/Oct/20

H o w m a n y s e t s o f 3 n u m b e r s e a c h c a n

b e f o r m e d f r o m t h e n u m b e r s {1, 2, 3, \dots, 20} i f n o

t w o c o n s e c u t i v e n u m b e r s a r e t o b e i n a s e t ?

Answered by bemath last updated on 19/Oct/20

\Rightarrow (\begin{matrix} 18 \\ 3 \end{matrix}) = \frac{18 \times 17 \times 16}{3 \times 2 \times 1} = 3 \times 17 \times 16 = 816

3 \times 17 \times 16

816.0

Commented by benjo_mathlover last updated on 19/Oct/20

y e s c o r r e c t

Commented by PRITHWISH SEN 2 last updated on 19/Oct/20

the problem is not that easy

Commented by PRITHWISH SEN 2 last updated on 19/Oct/20

1140 - (2 \times 17 \times 16 + 2 \times 2 \times 17) = 528

yes i think this will be the answer .

Answered by mr W last updated on 19/Oct/20

t o t a l : C_{3}^{20}

t h r e e c o n s e c u t i v e n u m b e r s :

1 + 2 + 3, 2 + 3 + 4, \dots, 18 + 19 + 20

\Rightarrow 18

t w o c o n s e c u t i v e n u m b e r s :

1 + 2 : 17

2 + 3 : 16

3 + 4 : 16

\dots

18 + 19 : 16

19 + 20 : 17

\Rightarrow 2 \times 17 + 17 \times 16 = 17 \times 18

t o t a l l y :

C_{3}^{20} - 18 - 17 \times 18 = 816

Commented by benjo_mathlover last updated on 20/Oct/20

y e s \dots

Commented by PRITHWISH SEN 2 last updated on 20/Oct/20

but sir {isn}^{'} t the set (1, 2) and (2, 1) both be consider as

the set of consecutive numbers ?

Commented by mr W last updated on 20/Oct/20

y e s . t h e y a r e e v e n t h e s a m e s e t!

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