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`

Prove that ^mC_1^n C_m-^m C_2^(2n)C_m+^m C_3^(3n)C_m-.....=(-1)^(m-1)n^mdot

Prove that mC_(1)^(n)C_(m)-^(m)C_(2)^(2n)C_(m)+^(m)C_(3)^(3n)C_(m)-...=(-1)^(m-1)n^(m)

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

Using binomial theorem (without using the formula for .^n C_r ) , prove that .^nC_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

Using binomial theorem (without using the formula for .^n C_r ) , prove that .^nC_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

Using binomial theorem (without using the formula for sim nC_(r)), prove that ^nC_(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_(4)

If m in N and mgeq2 prove that: |1 1 1\ ^m C_1\ ^(m+1)C_1\ ^(m+2)C_1\ ^m C_2\ ^(m+1)C_2\ ^(m+2)C_2|=1 .

If m in N and m>=2 prove that: |111^(m)C_(1)^(m+1)C_(1)^(m+2)C_(1)^(m)C_(2)^(m+1)C_(2)^(m+2)C_(2)|=1

Show that: ^mC_1+^(m+1)C_2+^(m+2)C_3+…….+^(m+n-1)C_n=^nC_1+^(n+1)C_2+^(n+2)C_3+…+^(n+m-1)C_m .

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