Q. Find the no. of non negative integral solutions of the eqn- a+b+c =13, where a,b and c are distinct? Ans is 84. Why can't we use the formula 13+3-1 C 3-1 ?

Because a,b,c are distinct. The formula doesn't handle that

Hi
n+r-1Cr-1 will have those cases as well where two of a,b and c are equal so we have to subtract them .
We get 15C2=105
Now 2m+n =13 so we get (0,0,13) (1,1,11) (2,2,9) (3,3,7) (4,4,5) (5,5,3) (6,6,1)
As these can be arranged in 3!/2! =3 ways so we get
105-7*3=84

You can use this formula... You have to subtract the value with the number of repetetions ... ex- (3,3,7), (4,4,5) etc.... Each case has 3 different values... So minus all the ( repetetion * 3)

how do you filter the cases when (without enumeration)?

Because a,b,c are distinct this formula wont be applicable here as it counts some cases where a,b and c are same too (eg. 3+3+7 is also counted where a and b are non-distinct)