Find the coefficient of `x^(25)` in expansion of expression `sum_(r=0)^(50)^(50)C_r(2x-3)^r(2-x)^(50-r)` .

Find the coefficient of x^(50) in the expansion of (1+x)^(101)xx(1-x+x^2)^(100)dot

Find the Co-efficient of x^(4) in the expansion of (1+x^(3))^(50)(x^(2)+1/x)^(5) .

The coefficient of x^(1274) in the expansion of (x+1)(x-2)^(2)(x+3)^(3)(x-4)^(4)…(x+49)^(49)(x-50)^(50) is

Find the sum sum_(r=0)^n^(n+r)C_r .

For r = 0, 1,"…..",10 , let A_(r),B_(r) , and C_(r) denote, respectively, the coefficient of x^(r ) in the expansion of (1+x)^(10), (1+x)^(20) and (1+x)^(30) . Then sum_(r=1)^(10) A_(r)(B_(10)B_(r ) - C_(10)A_(r )) is equal to

The coefficient of x^r[0lt=rlt=(n-1)] in lthe expansion of (x+3)^(n-1)+(x+3)^(n-2)(x+2)+(x+3)^(n-3)(x+2)^2++(x+2)^(n-1) is ^n C_r(3^r-2^n) b. ^n C_r(3^(n-r)-2^(n-r)) c. ^n C_r(3^r+2^(n-r)) d. none of these

The coefficient of x^(50) in (x+^(101)C_(0))(x+^(101)C_(1)).....(x+^(101)C_(50)) is

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

Find the sum sum_(i=0)^r.^(n_1)C_(r-i) .^(n_2)C_i .