Results Related to Arrangement:

(i) ⁿC_r = ⁿC_n-r

(a) If ⁿC_r = ⁿC_k ,  then r = k or n-r =k

(b) ⁿC_r + ⁿC_r-1 = ⁿ⁺¹C_r

(c) ⁿC_r = n/r ^n-1C_r-1

(v)

(vi)  (a) If n is even , ⁿC_r  is greatest for  r = n/2

            (b) If n is odd, ⁿC_r  is greatest for r = 

Example:      (a)       How many diagonals are there in an n-sided polygon (n> 3).

                    (b)       How many triangles can be formed by joining the vertices of an n- sided polygon. How many of these triangles have

                                          (i)      exactly one side common with that of the polygon

                                          (ii)     exactly two sides common with that of the polygon

                                          (iii)    no sides common with that of the polygon

Solution:       (a) The number of lines formed by joining the n vertices of a polygon

= number of selections of 2 points from the given n points 

            Out of  ⁿC₂ lines , n lines are the sides of the polygon.

                        Hence the number of diagonals = ⁿC₂ -n

(b) Number of triangles formed by joining the vertices of the polygon = number of selections of 3 points from n points.

                        Let the vertices of the polygon be marked as A₁, A₂,A₃,------A_n.

(i) Select two consecutive vertices A₁, A₂ of  the polygon. For the required triangle, we can select the third vertex from the points A₄,A₅, -----A_n-1 . This can be done in  ^n-4C₁ ways. Also two consecutive points (end points of a side of polygon) can be selected in n ways. Hence the total number of required triangles =n. ^n-4C₁ = n(n-4).

(ii) For the required triangle, we have to select three consecutive vertices of the polygon. i.e. (A₁ A₂ A₃), (A₂ A₃ A₄), (A₃ A₄ A₅),----------- ,(A_n A₁ A₂). This can be done in n ways.

(iii)Triangles having no side common + triangles having exactly one side common + triangles having exactly two sides common (with those of the polygon) = Total number of triangles formed

                        => Triangles having no side common with those of the polygon

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