# Number of ways to select three items from $n$ items

Number of ways to select three items from $$n$$ distinct items are :- $$\binom{n}{3}$$

Modified version of the problem :- Given $$n$$ boxes, each box having a particular amount of items , what is the number of ways to select three items such that each of the three items belong to different boxes ?

Example:- Let $$n=3$$,

First box contains $$x1=2$$ items.

Second box contains $$x2=3$$ items.

Third box contains $$x3=1$$ item(s).

Hence, total number of ways to select three items :- $$6$$(ways).

For any given $$n$$ and given $$x1,x2,x3.......$$till $$n^th$$ index, what are the number of ways to select three items such that each item belongs to a different box ?

$$\displaystyle \large \sum_{a=1}^{n-2}\sum_{b=a+1}^{n-1}\sum_{c=b+1}^{n}x_ax_bx_c$$.
The triple summation is something constructed to traverse through each possible combination of three boxes without repetition, while $$x_ax_bx_c$$ multiplies the number of items in the three boxes for each combination.