Prove that P(10,3) = P(9,3) + 3P(9,2)

Let P(n) be the statement " n^2-n+41 " is prime. Prove that P(1), P(2) and P(3) are true. Also prove that P(41) is not true. How does this not contradict the Principle of Induction ?

If P (n) is the statement : "2^(3n)- 1 is an integral multiple of 7”, prove that P(1), P(2) and P (3) are true.

If P (n) is the statement : "n (n + 1) (n + 2) is divisible by 12”, prove that P(3) and P(4) are true, but P(5) is not true.

Prove that (9!)/((9 - 3)!) = 504

Prove that : ^10P_3= "^9P_3+3^9P_2 .

Prove that 1+1* ""^(1)P_(1)+2* ""^(2)P_(2)+3* ""^(3)P_(3) + … +n* ""^(n)P_(n)=""^(n+1)P_(n+1).

If and B are two independent events with P (A) = (3)/(5) and P (B) = (4)/(9) then P (A cap B ) equals

If A and B are independent events such that P (A) =p, P (B)=2 p and P (Exactly one of A, B)= (5)/(9) then find p

Given that P(3,2,-4), Q(5,4,-6) and R(9,8,-10) are collinear. Find the ratio in which Q divides PR.

If A and B are two events such that P (A |B) = p , P (A)=p, P (B) = (1)/(3) and p (A cup B) = (5)/(9) then p = …….. .