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`

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 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

The value of determinant |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| is equal to 1 b. -1 c. 0 d. none of these

If m, n, r, in N then .^(m)C_(0).^(n)C_(r) + .^(m)C_(1).^(n)C_(r-1)+"…….."+.^(m)C_(r).^(n)C_(0) = coefficient of x^(r) in (1+x)^(m)(1+x)^(n) = coefficient of x^(f) in (1+x)^(m+n) The value of r for which S = .^(20)C_(r.).^(10)C_(0)+.^(20)C_(r-1).^(10)C_(1)+"........".^(20)C_(0).^(10)C_(r) is maximum can not be

If m, n, r, in N then .^(m)C_(0).^(n)C_(r) + .^(m)C_(1).^(n)C_(r-1)+"…….."+.^(m)C_(r).^(n)C_(0) = coefficient of x^(r) in (1+x)^(m)(1+x)^(n) = coefficient of x^(f) in (1+x)^(m+n) The value of r(0 le r le 30) for which S = .^(20)C_(r).^(10)C_(0) + .^(20)C_(r-1).^(10)C_(1) + ........ + .^(20)C_(0).^(10)C_(r) is minimum can not be

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

The value of ^n C_1+^(n+1)C_2+^(n+2)C_3++^(n+m-1)C_m is equal to (a)^m+n C_(n-1) (b)^m+n C_(n-1) (c)^mC_(1)+^(m+1)C_2+^(m+2)C_3++^(m+n-1) (d)^m+1C_(m-1)