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Question Number 118694 by benjo_mathlover last updated on 19/Oct/20
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 ?
$${How}\:{many}\:{sets}\:{of}\:\mathrm{3}\:{numbers}\:{each}\:{can} \\ $$$${be}\:{formed}\:{from}\:{the}\:{numbers}\:\left\{\:\mathrm{1},\mathrm{2},\mathrm{3},…,\mathrm{20}\:\right\}\:{if}\:{no} \\ $$$${two}\:{consecutive}\:{numbers}\:{are}\:{to}\:{be}\:{in}\:{a}\:{set}\:? \\ $$
Answered by bemath last updated on 19/Oct/20
⇒  (((18)),((  3)) ) = ((18×17×16)/(3×2×1)) = 3×17×16=816  3×17×16  816.0
$$\Rightarrow\:\begin{pmatrix}{\mathrm{18}}\\{\:\:\mathrm{3}}\end{pmatrix}\:=\:\frac{\mathrm{18}×\mathrm{17}×\mathrm{16}}{\mathrm{3}×\mathrm{2}×\mathrm{1}}\:=\:\mathrm{3}×\mathrm{17}×\mathrm{16}=\mathrm{816} \\ $$$$\mathrm{3}×\mathrm{17}×\mathrm{16} \\ $$$$\mathrm{816}.\mathrm{0} \\ $$
Commented by benjo_mathlover last updated on 19/Oct/20
yes correct
$${yes}\:{correct} \\ $$
Commented by PRITHWISH SEN 2 last updated on 19/Oct/20
the problem is not that easy
$$\mathrm{the}\:\mathrm{problem}\:\mathrm{is}\:\mathrm{not}\:\mathrm{that}\:\mathrm{easy} \\ $$
Commented by PRITHWISH SEN 2 last updated on 19/Oct/20
1140−(2×17×16+2×2×17)=528   yes i think this will be the answer.
$$\mathrm{1140}−\left(\mathrm{2}×\mathrm{17}×\mathrm{16}+\mathrm{2}×\mathrm{2}×\mathrm{17}\right)=\mathrm{528}\: \\ $$$$\mathrm{yes}\:\mathrm{i}\:\mathrm{think}\:\mathrm{this}\:\mathrm{will}\:\mathrm{be}\:\mathrm{the}\:\mathrm{answer}. \\ $$
Answered by mr W last updated on 19/Oct/20
total: C_3 ^(20)     three consecutive numbers:   1+2+3, 2+3+4,...,18+19+20  ⇒18    two consecutive numbers:   1+2: 17  2+3: 16  3+4: 16  ...  18+19: 16  19+20: 17  ⇒2×17+17×16=17×18    totally:  C_3 ^(20) −18−17×18=816
$${total}:\:{C}_{\mathrm{3}} ^{\mathrm{20}} \\ $$$$ \\ $$$${three}\:{consecutive}\:{numbers}:\: \\ $$$$\mathrm{1}+\mathrm{2}+\mathrm{3},\:\mathrm{2}+\mathrm{3}+\mathrm{4},…,\mathrm{18}+\mathrm{19}+\mathrm{20} \\ $$$$\Rightarrow\mathrm{18} \\ $$$$ \\ $$$${two}\:{consecutive}\:{numbers}:\: \\ $$$$\mathrm{1}+\mathrm{2}:\:\mathrm{17} \\ $$$$\mathrm{2}+\mathrm{3}:\:\mathrm{16} \\ $$$$\mathrm{3}+\mathrm{4}:\:\mathrm{16} \\ $$$$… \\ $$$$\mathrm{18}+\mathrm{19}:\:\mathrm{16} \\ $$$$\mathrm{19}+\mathrm{20}:\:\mathrm{17} \\ $$$$\Rightarrow\mathrm{2}×\mathrm{17}+\mathrm{17}×\mathrm{16}=\mathrm{17}×\mathrm{18} \\ $$$$ \\ $$$${totally}: \\ $$$${C}_{\mathrm{3}} ^{\mathrm{20}} −\mathrm{18}−\mathrm{17}×\mathrm{18}=\mathrm{816} \\ $$
Commented by benjo_mathlover last updated on 20/Oct/20
yes...
$${yes}… \\ $$
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 ?
$$\mathrm{but}\:\mathrm{sir}\:\mathrm{isn}'\mathrm{t}\:\mathrm{the}\:\mathrm{set}\:\left(\mathrm{1},\mathrm{2}\right)\:\mathrm{and}\:\left(\mathrm{2},\mathrm{1}\right)\:\mathrm{both}\:\mathrm{be}\:\mathrm{consider}\:\mathrm{as} \\ $$$$\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{consecutive}\:\mathrm{numbers}\:? \\ $$
Commented by mr W last updated on 20/Oct/20
yes. they are even the same set!
$${yes}.\:{they}\:{are}\:{even}\:{the}\:{same}\:{set}! \\ $$

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