Combinatorial numbers. For ex:- State the conditions when nCr that is n!/r!*(n-r)! is a triangular number that means in the form of n(n+1)/2 . An Example; 6C2 is 15 which can be expressed in the form of n(n+1)/2 where n in this case comes 5 and so (5 x 6)/2 comes out to be 15.
There are two other case such as 10C3 which comes out to be 120 and which can also be expressed in the form of a triangular number n(n+1)/2 where n comes out to be 15 as (15 x 16)/2 comes out to be 120 only. But as in case (n)C(n-1). It comes out to be traingular no. only when n is itself a triangular no. EX: 6C5 is 6 so 6 is a triangular no. as 6 is itself which is n is a triangular no. Second case where n is not a triangular number. Ex: 17C16 is 17 which is not a triangular no. as n itself that is 17 is not a triangular number.
So state the conditions when is nCr a triangular number.