Q)Choose at least some members frome the set A={14,15,...,20,22,23,...,28} so that in includes three consecutive members?


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Question Number 209746 by MM42 last updated on 20/Jul/24

$Q) C h o o s e a t l e a s t s o m e m e m b e r s$ $f r o m e t h e s e t A = {14, 15, . . ., 20, 22, 23, . . ., 28}$ $s o t h a t i n i n c l u d e s t h r e e c o n s e c u t i v e$ $m e m b e r s ?$
Commented by MM42 last updated on 20/Jul/24

$f o r e x a m p l e n = s e l e c t n u m b e r s$ $n = 3; 14, 15, 16 ✓ o r 14, 15, 17 \times \Rightarrow n o t t r u e$ $n = 4; 17, 22, 23, 24 ✓ o r 15, 17, 18, 20 \times \Rightarrow n o t t r u e$ $n = 6; 15, 17, 18, 19, 20, 27 ✓ o r 18, 19, 22, 24, 25, 28 \times \Rightarrow n o t t r u e$ $m i n (n) = ?; a_{1}, a_{2}, a_{3}, . . ., a_{n}$ $\Rightarrow \exists 1 < i, j, k < n; a_{k} = a_{j} + 1 = a_{i} + 2$
	Answered by MM42 last updated on 20/Jul/24

	$T h e l e a s t n u m b e r o f c h o i c e s c a n b e 11 .$ $b e c a u s e i n t h e s e t {14, 15, 17, 18, 20, 22, 24, 25, 27, 28} t h e r e a r e n o t h r e e$ $c o n s e c u t i v e n u m b e r s .$ $i f w e p u t a n y o t h e r n u m b e r i n t h e s e t, w e w i l l h a v e$ $t h r e e c o n s e c u t i v e n u m b e r s i n t h e c o l l e c t i o n . ★$
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