A probability problem by Achal Jain

Let the general case be:

$\displaystyle \Rightarrow \quad \quad x_1 + x_2 + \ldots + x_r = n$

$\displaystyle \text{where } n \in N, x_1,x_2,\ldots,x_r \text{ are non-negative integers.}$

This problem can be viewed as:

Suppose we have $\displaystyle n$ objects to begin with. We want to separate them in $\displaystyle r$ groups. We do so by using $\displaystyle r-1$ sticks to segregate the objects.(Assuming the $\displaystyle n$ objects are arranged in a straight line.)

The number of objects to the right of every stick (until we reach the next stick, if any) will be the number of objects in that group.

We are required to insert sticks in the gaps between objects such that a group has whole number of objects.

There are $\displaystyle n$ gaps to the right of the $\displaystyle n$ objects to place the sticks. And an additional $\displaystyle r-1$ gaps are assumed to be present at the left for the case when one group has all the objects and others have none.

Thus there are $\displaystyle r-1$ gaps to be selected from a total of $\displaystyle n+r-1$ gaps.

The total number of ways in which this can be done $\displaystyle = { n+r-1 \choose r-1}$

Substituting, $\displaystyle n = 10$ and $\displaystyle r = 4$ , we get $\displaystyle \boxed{286}$ .

Partition method...