How many non – negative integral solutions are possible for the given equation?
x + y + z = 16
Ans: 153
x + y + z = 16
Ans: 153
Let's say we have to divide 16 prizes among 3 boys such that each boy gets at least one prize i.e x+y+z =16, where x, y and z are positive integers.
Imagine 16 prizes placed one besides the other. Now we have 15 gaps in between them and to divide this into 3 parts we nee d to select any 2 two gaps from the available 15 gaps which can be done in 15C2 ways = 105 ways.
Now in the given question x, y and z are non-negative which mean a boy can recieve any number of prizes i.e 0 is also allowed. Imagining as in the above case we will have 18 (15+3)gaps from which we need to select 2 gaps which can be done in 18C2 = 153 ways.
So in general....
The number of solutions of the equation x1+x2+--------+xr = n, where x1,x2,...xr belongs to positive integers = (n-1)C(r-1)
The number of solutions of the equation x1+x2+--------+xr = n, where x1,x2,...xr belongs to non-negative integers = (n+r-1)C(r-1)
