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 ?


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

$H o w m a n y s e t s o f 3 n u m b e r s e a c h c a n$ $b e f o r m e d f r o m t h e n u m b e r s {1, 2, 3, . . ., 20} i f n o$ $t w o c o n s e c u t i v e n u m b e r s a r e t o b e i n a s e t ?$
	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

	$t o t a l : C_{3}^{20}$ $t h r e e c o n s e c u t i v e n u m b e r s :$ $1 + 2 + 3, 2 + 3 + 4, . . ., 18 + 19 + 20$ $\Rightarrow 18$ $t w o c o n s e c u t i v e n u m b e r s :$ $1 + 2 : 17$ $2 + 3 : 16$ $3 + 4 : 16$ $. . .$ $18 + 19 : 16$ $19 + 20 : 17$ $\Rightarrow 2 \times 17 + 17 \times 16 = 17 \times 18$ $t o t a l l y :$ $C_{3}^{20} - 18 - 17 \times 18 = 816$
	Commented by benjo_mathlover last updated on 20/Oct/20

	$y e s . . .$
	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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