If `m >1,n in N` show that `1^m+2^m+2^(2m)+2^(3m)+....+2^(n m-m)> n^(1-m)(2^n-1)^mdot`

Prove that ^m C_1^n C_m-^m C_2^(2n)C_m+^m C_3^(3n)C_m-=(-1)^(m-1)n^mdot

A function f is defined by f(x)=|x|^m|x-1|^nAAx in Rdot The local maximum value of the function is (m ,n in N), 1 (b) m^nn^m (m^m n^n)/((m+n)^(m+n)) (d) ((m n)^(m n))/((m+n)^(m+n))

Prove that sum_(r=1)^(m-1)(2r^2-r(m-2)+1)/((m-r)^m C_r)=m-1/mdot

Solve the any one of following sub questions: If m and n are two distinct numbers then prove that m^(2) - n^(2), 2mn and m^(2) + n^(2) is a pythagorean triplet.

For any positive integer (m,n) (with ngeqm ), Let ((n),(m)) =.^nC_m Prove that ((n),(m)) + 2((n-1),(m))+3((n-2),(m))+....+(n-m+1)((m),(m))

ABCD is a square of length a, a in N , a > 1. Let L_1, L_2 , L_3... be points on BC such that BL_1 = L_1 L_2 = L_2 L_3 = .... 1 and M_1,M_2 , M_3,.... be points on CD such that CM_1 = M_1M_2= M_2 M_3=... = 1 . Then sum_(n = 1)^(a-1) ((AL_n)^2 + (L_n M_n)^2) is equal to :

Using binomial theorem (without using the formula for ^n C_r ) , prove that "^n C_4+^m C_2-^m C_1^n C_2 = ^m C_4-^(m+n)C_1^m C_3+^(m+n)C_2^m C_2-^(m+n)C_3^m C_1+^ (m+n)C_4dot

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

In a game a coin is tossed 2n+m times and a player wins if he does not get any two consecutive outcomes same for at least 2n times in a row. The probability that player wins the game is a. (m+2)/(2^(2n)+1) b. (2n+2)/(2^(2n)) c. (2n+2)/(2^(2n+1)) d. (m+2)/(2^(2n))