`a+ar+ar^(2)+......+ar^(n-1)=(a(r^(n)-1))/(r-1)`

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

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

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

the roots of the equations |{:(.^(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

Prove that if 1 le r le n " then " n xx^((n-1))C_(r-1)= (n-r+1).^(n)C_(r-1)

If (1+x)^n=sum_(r=0)^n^n C_r , show that C_0+(C_1)/2++(C_n)/(n+1)=(2^(n+1)-1)/(n+1) .

A car is parked among N cars standing in a row, but not at either end. On his return, the owner finds that exactly "r" of the N places are still occupied. The probability that the places neighboring his car are empty is a. ((r-1)!)/((N-1)!) b. ((r-1)!(N-r)!)/((N-1)!) c. ((N-r)(N-r-1))/((N-1)(N+2)) d. "^((N-r)C_2)/(.^(N-1)C_2)

If a_1,a_2,a_3…a_n are in H.P and f(k)=(Sigma_(r=1)^(n) a_r)-a_k then a_1/f(1),a_2/f(3),….,a_n/f(n) are in

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