Prove that ` sum_(r=1)^5 ^5C_r=31 `

Prove that sum_(r=0)^(5) ""^(5)C_(r )=32 .

If D_r=|2^(r-1)2(3^(r-1))4(5^(r-1))x y z2^n-1 3^n-1 5^n-1| then prove that sum_(r=1)^n D_r=0.

Prove that sum_(r=1)^(10) = sin^(2) rpi/18 =5 .

Prove that sum_(r=1)^(n)(-1)^(r-1)(1+(1)/(2)+(1)/(3)+...+(1)/(r))^(n)C_(r)=(1)/(n)

If (1+x)^n=sum_(r=0)^n C_rx^r then prove that sum_(r=0)^n (C_r)/((r+1)2^(r+1))=(3^(n+1)-2^(n+1))/((n+1)2^(n+1))

Prove that sum_(n)^(r=0) ""^(n)C_(r)*3^(r)=4^(n).

Prove that sum_(r=0)^(n)C_(r)sin rx cos(n-r)x=2^(n-1)sin(nx)

sum_(r=1)^(6)6C_(r)

Prove that sum_(r=1)^(m-1)(2r^(2)-r(m-2)+1)/((m-r)^(m)C_(r))=-(1)/(m)

Prove that sum_(r=0)^(s)sum_(s=1)^(n)C_(s)^(n)C_(r)=3^(n)-1