Prove that `nC_r+3.nC_(r-1)+3nC_(r-2)+nC_(r-3)=(n+3)C_r`

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

If 2lerlen show that ^nC_r +2 ^nC_(r-1) +^nC_(r-2)=^(n+2)C_r

show that ^nC_r+ ^(n-1)C_(r-1)+ ^(n-1)C_(r-2)= ^(n+1)C_r

Prove that .^(n)C_(0) - .^(n)C_(1) + .^(n)C_(2) - .^(n)C_(3) + "……" + (-1)^(r) .^(n)C_(r) + "……" = (-1)^(r ) xx .^(n-1)C_(r ) .

Prove that "^n C_r+^(n-1)C_r+...+^r C_r=^(n+1)C_(r+1) .

Prove that r_1+r_2+r_3-r=4R

Prove that ^nC_0 "^(2n)C_n-^nC_1 ^(2n-2)C_n +^nC_2 ^(2n-4)C_n =2^n

If n is a positive integer then using the indentiy (1+x)^(n)=(1+x)^(3)(1+x)^(n-3) , prove that ""^(n)C_(r)=""^(n-2)C_(r)+3*""^(n-3)C_(r-1)*""^(n-3)C_(r-2)+""^(n-3)C_(r-3)

Let A B C be a triangle with incenter I and inradius rdot Let D ,E ,a n dF be the feet of the perpendiculars from I to the sides B C ,C A ,a n dA B , respectively. If r_1,r_2a n dr_3 are the radii of circles inscribed in the quadrilaterals A F I E ,B D I F ,a n dC E I D , respectively, prove that (r_1)/(r-1_1)+(r_2)/(r-r_2)+(r_3)/(r-r_3)=(r_1r_2r_3)/((r-r_1)(r-r_2)(r-r_3))

Prove that sum_(r=1)^(k) (-3)^(r-1) (3n)^C_(2r-1) = 0 , where k = 3n//2 and n is an even integer.