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Question Number 228273 by AgniMath last updated on 04/Apr/26
An insect moves from (0,0) to (6,3) by moving through lattice points ((x,y) integers) moving either one unit right or one unit up at each step. Let N be the number of paths in which the line segments joining (2,1),(2,2),(3,2) and (4,2) are avoided. find N.
An insect moves from (0,0) to (6,3) by moving through lattice points ((x,y) integers) moving either one unit right or one unit up at each step. Let N be the number of paths in which the line segments joining (2,1),(2,2),(3,2) and (4,2) are avoided. find N.
Answered by mr W last updated on 06/Apr/26
Commented by mr W last updated on 06/Apr/26
N=45
$${N}=\mathrm{45} \\ $$
Commented by Tinku Tara last updated on 06/Apr/26
Total 9 steps needed 3U,6R and total possible path are = ((9),(3) ) = 84 R_1 =(2,1)→(2,2) R_2 =(2,2)→(3,2) R_3 =(3,2)→(4,2) From 84 we need to subtract route that go thru any of R_1 , R_2 , R_3 path thru R_1 =(0,0)→(2,1)→R_1 →(2,2)→(6,3) =3×5=15 (a) path thru R_2 =(0,0)→(2,2)→R_2 →(3,2)→(6,3) =6×4=24 (b) path thru R_3 =(0,0)→(3,2)→R_3 →(4,2)→(6,3) =10×3=30 (c) path thru R_1 +R_2 =3×4=12 (d) path thru R_2 +R_3 =6×3=18 (e) path thru R_1 +R_3 =3×1×3=9 (f) path thru R_1 +R_2 +R_3 =3×3=9 (g) path covering at least one of R_(1,) R_2 ,R_3 a+b+c−d−e−f+g =39 Answer: 84−39=45 n(A∪B∪C) formula is used
$$\mathrm{Total}\:\mathrm{9}\:\mathrm{steps}\:\mathrm{needed}\:\mathrm{3U},\mathrm{6R}\:\mathrm{and}\: \\ $$$$\mathrm{total}\:\mathrm{possible}\:\mathrm{path}\:\mathrm{are}\:=\:\begin{pmatrix}{\mathrm{9}}\\{\mathrm{3}}\end{pmatrix}\:=\:\mathrm{84} \\ $$$$\mathrm{R}_{\mathrm{1}} =\left(\mathrm{2},\mathrm{1}\right)\rightarrow\left(\mathrm{2},\mathrm{2}\right) \\ $$$$\mathrm{R}_{\mathrm{2}} =\left(\mathrm{2},\mathrm{2}\right)\rightarrow\left(\mathrm{3},\mathrm{2}\right) \\ $$$$\mathrm{R}_{\mathrm{3}} =\left(\mathrm{3},\mathrm{2}\right)\rightarrow\left(\mathrm{4},\mathrm{2}\right) \\ $$$$\mathrm{From}\:\mathrm{84}\:\mathrm{we}\:\mathrm{need}\:\mathrm{to}\:\mathrm{subtract} \\ $$$$\mathrm{route}\:\mathrm{that}\:\mathrm{go}\:\mathrm{thru}\:\mathrm{any}\:\mathrm{of}\:\mathrm{R}_{\mathrm{1}} ,\:\mathrm{R}_{\mathrm{2}} ,\:\mathrm{R}_{\mathrm{3}} \\ $$$$\mathrm{path}\:\mathrm{thru}\:\mathrm{R}_{\mathrm{1}} \\ $$$$\:\:\:\:\:\:\:=\left(\mathrm{0},\mathrm{0}\right)\rightarrow\left(\mathrm{2},\mathrm{1}\right)\rightarrow\mathrm{R}_{\mathrm{1}} \rightarrow\left(\mathrm{2},\mathrm{2}\right)\rightarrow\left(\mathrm{6},\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:=\mathrm{3}×\mathrm{5}=\mathrm{15}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({a}\right)\:\:\:\:\:\:\: \\ $$$$\mathrm{path}\:\mathrm{thru}\:\mathrm{R}_{\mathrm{2}} \\ $$$$\:\:\:\:\:\:\:=\left(\mathrm{0},\mathrm{0}\right)\rightarrow\left(\mathrm{2},\mathrm{2}\right)\rightarrow\mathrm{R}_{\mathrm{2}} \rightarrow\left(\mathrm{3},\mathrm{2}\right)\rightarrow\left(\mathrm{6},\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:=\mathrm{6}×\mathrm{4}=\mathrm{24}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\left({b}\right) \\ $$$$\mathrm{path}\:\mathrm{thru}\:\mathrm{R}_{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:=\left(\mathrm{0},\mathrm{0}\right)\rightarrow\left(\mathrm{3},\mathrm{2}\right)\rightarrow\mathrm{R}_{\mathrm{3}} \rightarrow\left(\mathrm{4},\mathrm{2}\right)\rightarrow\left(\mathrm{6},\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:=\mathrm{10}×\mathrm{3}=\mathrm{30}\:\:\:\:\:\:\:\:\:\:\:\:\left({c}\right) \\ $$$$\mathrm{path}\:\mathrm{thru}\:\mathrm{R}_{\mathrm{1}} +\mathrm{R}_{\mathrm{2}} =\mathrm{3}×\mathrm{4}=\mathrm{12}\:\left({d}\right) \\ $$$$\mathrm{path}\:\mathrm{thru}\:\mathrm{R}_{\mathrm{2}} +\mathrm{R}_{\mathrm{3}} =\mathrm{6}×\mathrm{3}=\mathrm{18}\:\left({e}\right) \\ $$$$\mathrm{path}\:\mathrm{thru}\:\mathrm{R}_{\mathrm{1}} +\mathrm{R}_{\mathrm{3}} =\mathrm{3}×\mathrm{1}×\mathrm{3}=\mathrm{9}\:\left({f}\right) \\ $$$$\mathrm{path}\:\mathrm{thru}\:\mathrm{R}_{\mathrm{1}} +\mathrm{R}_{\mathrm{2}} +\mathrm{R}_{\mathrm{3}} =\mathrm{3}×\mathrm{3}=\mathrm{9}\:\left({g}\right) \\ $$$$\mathrm{path}\:\mathrm{covering}\:\mathrm{at}\:\mathrm{least}\:\mathrm{one}\:\mathrm{of}\:\mathrm{R}_{\mathrm{1},} \mathrm{R}_{\mathrm{2}} ,\mathrm{R}_{\mathrm{3}} \\ $$$${a}+{b}+{c}−{d}−{e}−{f}+{g} \\ $$$$=\mathrm{39} \\ $$$$\mathrm{Answer}:\:\mathrm{84}−\mathrm{39}=\mathrm{45} \\ $$$${n}\left({A}\cup{B}\cup{C}\right)\:{formula}\:{is}\:{used} \\ $$

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