If `m` parallel lines in a plane are intersected by a family of `n` parallel lines, the number of parallelograms that can be formed is a. `1/4m n(m-1)(n-1)` b. `1/4m n(m-1)` c. `1/4m^2n^2` d. none of these

Assertion: If m parallel lines are intersected by n other parallel llines, then the number of parallelograms thus formed is (mn(m-1)(n-1))/4 , Reason: A selection of 4 lines 2 from m parallel lines and 2 from n parallel lines gives one parallelogram. (A) Both A and R are true and R is the correct explanation of A (B) Both A and R are true R is not the correct explanation of A (C) A is true but R is false. (D) A is false but R is true.

Straight lines are drawn by joining m points on a straight line of n points on another line. Then excluding the given points, the number of point of intersections of the lines drawn is (no tow lines drawn are parallel and no these lines are concurrent). a. 4m n(m-1)(n-1) b. 1/2m n(m-1)(n-1) c. 1/2m^2n^2 d. 1/4m^2n^2

A rectangle with sides of lengths (2n-1) and (2m-1) units is divided into squares of unit length. The number of rectangles which can be formed with sides of odd length, is (a) m^2n^2 (b) mn(m+1)(n+1) (c) 4^(m+n-1) (d) non of these

m points on one straight line are joined to n points on another straight line. The number of points of intersection of the line segments thus formed is (A) ^mC-2.^nC_2 (B) (mn(m-1)(n-1))/4 (C) (^mC_2.^nC_2)/2 (D) ^mC_2+^nC_2

If m ,n in Na n dm^2-n^2=13 ,t h e n(m+1)(n+1) is equal to a. 42 b. 56 c. 50 d. none of these

A B C is a right-angled triangle in which /_B=90^0 and B C=adot If n points L_1, L_2, ,L_nonA B is divided in n+1 equal parts and L_1M_1, L_2M_2, ,L_n M_n are line segments parallel to B Ca n dM_1, M_2, ,M_n are on A C , then the sum of the lengths of L_1M_1, L_2M_2, ,L_n M_n is (a(n+1))/2 b. (a(n-1))/2 c. (a n)/2 d. none of these

sum_(m=1)^(n) tan^(-1) ((2m)/(m^(4) + m^(2) + 2)) is equal to

If x and y are positive real numbers and m, n are any positive integers, then prove that (x^n y^m)/((1+x^(2n))(1+y^(2m))) lt 1/4

A is a set containing n elements. A subset P_1 is chosen and A is reconstructed by replacing the elements of P_1 . The same process is repeated for subsets P_1,P_2,....,P_m with m>1 . The number of ways of choosing P_1,P_2,....,P_m so that P_1 cup P_2 cup....cup P_m=A is (a) (2^m-1)^(mn) (b) (2^n-1)^m (c) (m+n)C_m (d) none of these

If \ ^m C_1=\ \ ^n C_2 then which is correct a. 2m=n b. 2m=n(n+1) c. 2m=(n-1) d. 2n=m(m-1)