The number of ways of arranging `m` positive and `n(lt m+1)` negative signs in a row so that no two are together is a.= `^m+1p_n` b.=`^n+1p_m`= c.=`^m+1c_n`="" d.=`^n+1c_m`

2m white counters and 2n red counters are arranged in a straight line with (m+n) counters on each side of central mark. The number of ways of arranging the counters, so that the arrangements are symmetrical with respect to the central mark is (A) .^(m+n)C_m (B) .^(2m+2n)C_(2m) (C) 1/2 ((m+n)!)/(m! n!) (D) None of these

Show that the numbers of permutations of n different things taken all at a time in which m particular things are never together is (n-m)!(n-m+1)! .

If the sum of m terms of an A.P. is the same as the sum of its n terms, then the sum of its (m+n) terms is (a). m n (b). -m n (c). 1//m n (d). 0

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

If nge3 is an integer prove that 2n+1 lt 2^n by P.M.I.

The ratio of the sum of m and n terms of an A.P. is m^(2) :n^(2) . Show that the ratio mth and nth term is (2m-1) : (2n-1).

If the sum of 1st M terms of an AP is m. and the sum of 1st N terms is n, show that- the 1st term of this AP is (mN(N-1)-nM(M-1))/(MN(N-M))

The number of ordered pairs (m,n) where m , n in {1,2,3,…,50} , such that 6^(m)+9^(n) is a multiple of 5 is

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

The ratio of the sums of m and n terms of an A.P. is m^2 : n^2. Show that the ratio of m^(th) and n^(th) term is (2m-1) : (2n -1) .