If `(1+x)^n = C_0 + C_1x + C_2x^2 + ……..+C_n.x^n` then find `C_0 - C_2 + C_4 - C_6 + …….`

If (1+x)^n = C_0 + C_1x + C_2x^2 + ……..+C_n.x^n then find C_1 - C_3 + C_5 + ……

If (1 + x + x^2)^n = C_0 + C_1x + C_2x^2 + C_3x^3 + ……..C_nx^n then find C_0 + C_3 + C_6 + …….

If (1 + x)^n = C_0+C_1x+C_2x^2 + ... + C_nx^n ,then for n odd, C_0^2-C_1^2+C_2^2-C_3^2 + ... +(-1) C_n^2 , is equal to

If (1+x)^(n) = C_(0) + C_(1)x + C_(2)x^(2) + "….." + C_(n)x^(n) , then C_(0) - (C_(0) + C_(1)) +(C_(0) + C_(1) + C_(2)) - (C_(0) + C_(1) + C_(2) + C_(3))+ "….." (-1)^(n-1) (C_(0) + C_(1) + "……" + C_(n-1)) is (where n is even integer and C_(r) = .^(n)C_(r) )

If (1 + x)^(n) = C_(0) + C_(1)x + C_(2) x^(2) + …+ C_(n) x^(n) , then for n odd, C_(1)^(2) + C_(3)^(2) + C_(5)^(2) +....+ C_(n)^(2) is equal to

If (1+x)^n=C_0+C_1x+C_2x^2+……..+C_nx^n , show that C_0/1.2-C_1/2.3+C_2/3.4-C_3/4.5+………..+(-1)^n C_n/((n+1)(n+2))=1/(n+2)

If (1 + x)^(n) = C_(0) + C_(1) x C_(2) x^(2) +…+ C_(n) x^(n) , then the sum C_(0) + (C_(0)+C_(1))+…+(C_(0) +C_(1) +…+C_(n -1)) is equal to

If (1 + x + x^2)^n = (C_0 + C_1x + C_2 x^2 + ............) then the value of C_0 C_1-C_1C_2+C_2C_3.....

If (1+ x)^(n) = C_(0) + C_(1) x + C_(2)x^(2) + ...+ C_(n)x^(n) , prove that C_(1) + 2C_(2) + 3C_(3) + ...+ n""C_(n) = n*2^(n-1)

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + …+ C_(n) x^(n) , prove that C_(0)^(2) - C_(1)^(2) + C_(2)^(2) -…+ (-1)^(n) *C_(n)^(2)= 0 or (-1)^(n//2) * (n!)/((n//2)! (n//2)!) , according as n is odd or even Also , evaluate C_(0)^(2) - C_(1)^(2) + C_(2)^(2) - ...+ (-1)^(n) *C_(n)^(2) for n = 10 and n= 11 .