Lesson Math circle level problem on Combinations
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<H2>Math circle level problem on Combinations</H2> <H3>Problem 1</H3>Seventeen lines drawn in a plane, with no 3 concurrent and no 2 parallel, divide the plane into closed regions (bounded on all sides) and open regions. What is the number of closed regions ? <B>Solution</B> <pre> Each group of 3 lines makes one triangle; and vise versa, each such a triangle defines a group of 3 lines by an unique way. So, there are {{{C[17]^3}}} triangles. Each group of 4 lines makes one quadrilateral; and vise versa, each such a quadrilateral defines a group of 4 lines by an unique way. So, there are {{{C[17]^4}}} quadrilaterals. Each group of 5 lines makes one pentagon; and vise versa, each such a pentagon defines a group of 5 lines by an unique way. So, there are {{{C[17]^5}}} pentagons. . . . . . . . . and so on . . . . . . Each group of 16 lines makes one 16-gon; and vise versa, each such a 16-gon defines a group of 16 lines by an unique way. So, there are {{{C[17]^16}}} 16-gons. Finally, all 17 lines together make 1 = {{{C[17]^17}}} 17-gon. So, the number of all closed regions is the sum R = {{{C[17]^3}}} + {{{C[17]^4}}} + {{{C[17]^5}}} + . . . {{{C[17]^16}}} + {{{C[17]^17}}}. If you complement this sum with the terms {{{C[17]^0}}} + {{{C[17]^1}}} + C{{{17]^2}}}, you will get R + {{{C[17]^0}}} + {{{C[17]^1}}} + C{{{17]^2}}} = {{{C[17]^0}}} + {{{C[17]^1}}} + C{{{17]^2}}} + {{{C[17]^3}}} + {{{C[17]^4}}} + {{{C[17]^5}}} + . . . {{{C[17]^16}}} + {{{C[17]^17}}}. The long sum in the right side is equal to {{{2^17}}}. Therefore, the number S under the question is equal to R = {{{2^17}}} - ({{{C[17]^0}}} + {{{C[17]^1}}} + C{{{17]^2}}}) = {{{2^17}}} - 1 - 17 - {{{(17*16)/2}}} = {{{2^17}}} - 1 - 17 - 136 = {{{2^17}}} - 154. </pre> My lessons on Permutations and Combinations in this site are - <A HREF =http://www.algebra.com/algebra/homework/Permutations/Introduction-to-Permutations.lesson>Introduction to Permutations</A> - <A HREF =http://www.algebra.com/algebra/homework/Permutations/PROOF-of-the-formula-on-the-number-of-permutations.lesson>PROOF of the formula on the number of Permutations</A> - <A HREF =http://www.algebra.com/algebra/homework/Permutations/Problems-on-Permutations.lesson>Simple and simplest problems on permutations</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Special-type-permutations-problems.lesson>Special type permutations problems</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/How-many-different-permutations-may-exist-ubder-given-restrictions.lesson>Problems on Permutations with restrictions</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Math-circle-level-problem-on-Permutations.lesson>Math circle level problem on Permutations</A> - <A HREF =http://www.algebra.com/algebra/homework/Permutations/Introduction-to-Combinations-.lesson>Introduction to Combinations</A> - <A HREF =http://www.algebra.com/algebra/homework/Permutations/PROOF-of-the-formula-on-the-number-of-combinations.lesson>PROOF of the formula on the number of Combinations</A> - <A HREF =http://www.algebra.com/algebra/homework/Permutations/Problems-on-Combinations.lesson>Problems on Combinations</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Combinations-problems-with-restrictions.lesson>Problems on Combinations with restrictions</A> - Math circle level problem on Combinations (this lesson) - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Arranging-elements-of-sets-containing-undistinguishable-elements.lesson>Arranging elements of sets containing indistinguishable elements</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Persons-sitting-around-a-circular-table.lesson>Persons sitting around a cicular table</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Combinatoric-problems-for-entities-other-than-permutations-and-combinations.lesson>Combinatoric problems for entities other than permutations and combinations</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Fundamental-counting-principle-problems.lesson>Fundamental counting principle problems</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Miscellaneous-problems-on-permutations-combinations-and-other-combinatoric-entities.lesson>Miscellaneous problems on permutations, combinations and other combinatoric entities</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Some-twisted-combinatorics-problem.lesson>Some twisted combinatorics problem</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Inclusion-Exclusion-principle.lesson>Inclusion-Exclusion principle problems</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/DERANGEMENT-problems.lesson>Derangement problems</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/In-how-many-ways-N-distinguishable-objects--can-be-distributed-among-n-different-boxes.lesson>In how many ways N distinguishable objects can be distributed among n different boxes ?</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Stars-and-bars-method-for-Combinatorics-problems-2.lesson>Stars and bars method for Combinatorics problems</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Men-and-women-standing-in-line-.lesson>Men and women standing in line</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/One-combinatorial-Geometry-problem-solved-using-the-Euler-formula.lesson>One combinatorial Geometry problem solved using the Euler formula</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/Nice-recreational-problems-on-permutations.lesson>Nice recreational problems on permutations</A> - <A HREF =https://www.algebra.com/algebra/homework/Permutations/OVERVIEW-the-lessons-on-Permutations-and-Combinations.lesson>OVERVIEW of lessons on Permutations and Combinations</A> Use this file/link <A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A> to navigate over all topics and lessons of the online textbook ALGEBRA-II.
