`((100-100))/((100-100))=2`

The value of ((100),(0))((200),(150))+((100),(1))((200),(151))+......+((100),(50))((200),(200)) equals (where ((n),(r ))="^(n)C_(r) )

Prove that ^100 C_0^(100)C_2+^(100)C_2^(100)C_4+^(100)C_4^(100)C_6++^(100)C_(98)^(100)C_(100)=1/2[^(200)C_(98)-^(100)C_(49)]dot

Prove that int_0^(102)(x-1)(x-2)(x-100) x(1/((x-1)+1/((x-2))+1/((x-100))dx=101 !-100 !

If f(x)=(x-1)^(100)(x-2)^(2(99))(x-3)^(3(98))…(x-100)^(100), then the value of (f'(101))/(f(101)) is

Let t_(100)=sum_(r=0)^(100)(1)/(("^(100)C_(r ))^(5)) and S_(100)=sum_(r=0)^(100)(r )/(("^(100)C_(r ))^(5)) , then the value of (100t_(100))/(S_(100)) is (a) 1 (b) 2 (c) 3 (d) 4

The coefficient of x^(53) in the expansion sum_(m=0)^(100)^100C_m(x-3)^(100-m)2^m is (a) 100 C_(47) (b.) 100 C_(53) (c.) -100C_(53) (d.) none of these

Let f(x)=int_(0)^(x)(sin^(100)t)/(sin^(100)t+cos^(100)t)dt , then f(2pi)=

[-(1)/(3)]+[-(1)/(3)-(1)/(100)]+[-(1)/(3)-(2)/(100)]+….+[-(1)/(3)-(99)/(100)] is equal to (where [.] denotes greatest integer function)

If f(x) is a differntiable function satisfying the condition f(100x)=x+f(100x-100) , forall x in R and f(100)=1 , then f(10^(4)) is

Let S_(k) , where k = 1,2 ,....,100, denotes the sum of the infinite geometric series whose first term is (k -1)/(k!) and the common ratio is (1)/(k) . Then, the value of (100^(2))/(100!) +sum_(k=2)^(100) | (k^(2) - 3k +1) S_(k)| is....