Match the following:

	Column-I		Column-II
A	Number of triangles joining the vertices of the polygon having 35 diagonals is	P	1296
B	Maximum number of point of intersection of six circles in the plane, is	Q	57
C	Number of straight lines joining any two of 12 points of which 5 are collinear, is	R	120
D	Number of rectangles in a $8\times 8$ chess board, is	S	30

(A)We know that if polygon has n sides, then the total number of diagonals is ${ }^n{C}_2-n$ .
 ${\therefore }^n{C}_2-n=35\Rightarrow \frac{n(n-3)}{2}=35$
$\Rightarrow n^2-3n-70=0\Rightarrow (n-10)(n+7)=0$ 
 $\therefore n=10$
Now,with 10 vertices, number of triangles formed  $={}^{10}{C}_3=120$
(B)Number of circles=6
$\therefore$  Maximum number of point of intersection  $=2\times {}^6{C}_2=30$
(C) Number of straight lines with 12 points $={}^{12}{C}_2=66$
Number of straight lines with 5 points $={}^5{C}_2=10$
$\therefore$  Required number of straight lines  $=66-10+1=57$
(D)A  $8\times 8$  chess board having 9 horizontal lines and 9 vertical lines.
$\therefore$ Required number of rectangles formed
 $={}^9{C}_2\times {}^9{C}_2=1296$