n C r and n P r

Given that C(n, r) : C(n,r + 1) = 1 : 2 and C(n,r + 1) : C(n,r + 2) = 2 : 3 . What is P(n, r) : C(n, r) equal to ?

In P Q R , M and N are points on sides P Q and P R respectively such that P M=15 c m and N R=8c m . If P Q=25 c m and P R=20 c m state whether M N Q R .

If .^(n)P_(r)=.^(n)P_(r+1) and .^(n)C_(r)=.^(n)C_(r-1) , find n and r.

If ""^(n)P_r = ""^(n)P_(r+1) and ""^(n)C_r = ""^(n)C_(r-1) ,then the values of n and r are

If ""(n)C_(0), ""(n)C_(1), ""(n)C_(2), ...., ""(n)C_(n), denote the binomial coefficients in the expansion of (1 + x)^(n) and p + q =1 sum_(r=0)^(n) r^(2 " "^n)C_(r) p^(r) q^(n-r) = .

If ""^(n)C_(0), ""^(n)C_(1),..., ""^(n)C_(n) denote the binomial coefficients in the expansion of (1 + x)^(n) and p + q = 1 , then sum_(r=0)^(n) ""r.^(n)C_(r) p^(r) q^(n-r) =

Let n and r be no negative integers suych that r<=n. Then,^(n)C_(r)+^(n)C_(r-1)=^(n+1)C_(r)