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

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

Prove that ""^(n-1)P_r+r^(n-1)p_r-1=^nP_r dot""

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

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 ) .

.^((n-1)C_(r) + ^((n-1)) C_((r-1)) is

Prove that sum_(r=0)^(2n)r(.^(2n)C_r)^2=n^(4n)C_(2n) .

Prove ""^(n)P_(r)=""^(n-1)P_(r) +r. ""^(n-1)P_(r-1)

Prove that (r+1)^n C_r-r^n C_r+(r-1)^n C_2-^n C_3++(-1)^r^n C_r = (-1)^r^(n-2)C_rdot

The roots of the equation |^x C_r^(n-1)C_r^(n-1)C_(r-1)^(x+1)C_r^n C_r^n C_(r-1)^(x+2)C_r^(n+1)C_r^(n+1)C_(r-1)|=0 are x=n b. x=n+1 c. x=n-1 d. x=n-2

If (18 x^2+12 x+4)^n=a_0+a_(1x)+a2x2++a_(2n)x^(2n), prove that a_r=2^n3^r(^(2n)C_r+^n C_1^(2n-2)C_r+^n C_2^(2n-4)C_r+) .