Prove that `100C_0, 100C_2+100C_2, 100C_4+100C_4. 100C_6+.....+100C_98, 100C_100=1/2 [200C_98-100C_49]`

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 .^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 ^100C_(2)^(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)]

Evaluate: ^100C_97

Evaluate: ^100C_97

The value of "^100C_98 =

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)

Sum of the series (100C_(1))^(2)+2.(100C_(2))^(2)+3.(100C_(3))^(2)+....+100(100C_(100))^(2) equals

Convert: -100@ C to kelvin