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 the following identieties using the theory of permutation where C_(0),C_(1),C_(2),……C_(n) are the combinatorial coefficents in the expansion of (1+x)^n,n in N: ""^(100)C_(10)+5.""^(100)C_(11)+10 .""^(100)C_(12)+ 10.""^(100)C_(13)+ 10.""^(100)C_(14)+ 10.""^(100)C_(15)=""^(105)C_(90)

The value of ""^(40)C_(0) xx ""^(100)C_(40) _ ""^(40)C_(1) xx ""^(99)C_(40) + ""^(40)C_(2) xx ""^(98)C_(40) -"……." + ""^(40)C_(40) xx ""^(60)C_(40) is equal to "____" .

Evaluate: ^100C_97

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

If ""^(100)C_(6)+4." "^(100)C_(7)+6." "^(100)C_(8)+4." "^(100)C_(9)+""^(100)C_(10) has the value equal to " "^(x)C_(y) , then the possible value (s) of x+y can be :

If ^100 C_5+5^(100)C_6+10^(100)C_7+10^(100)C_8+5^(100)C_9+^(100)C_(10) has the value equal to ^n C_r , then least value of (n+r) is equal to 200 (2) 195 (3) 115 (4) 105

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

The value of sum_(r=0)^50 (.^(100)C_r.^(200)C_(150+r)) is equal to

(C_(0))/(1)+(C_(1))/(2)+(C_(2))/(3)+ . . . .+(C_(100))/(101) equals

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