Find the sum `^20C_10.^15C_0+^20C_9.^15C_1+^20C_8.^15C_2+....+^20C_0.^15C_10`

Find the sum sum_(k=0)^(10)2^(20)C_(k)

The sum of the series ""^20C_0+""^20C_1+""^20C_2+...+""^20C_9 is

The sum of the series .^(20)C_(0)-.^(20)C_(1)+ .^(20)C_(2)-.^(20)C_(3)+...-.+ .^(20)C_(10) is -

1.^(20)C_(1)-2.^(20)C_(2)+3.^(20)C_(3)-...-20.^(20)C_(20)

Show that: .^20C_13+^20C_14-^20C_6-^20C_7=0

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