In how many ways can a positive integer $n$ be expressed as a summation of positive integers less than $n$ For example if I take $n=5$, then I can express it in the following ways:


*

*$1+1+1+1+1$

*$2+3$

*$3+2$

*$1+4$

*$4+1$

*$1+1+3$

*$1+3+1$

*$3+1+1$

*$2+2+1$

*$2+1+2$

*$1+2+2$


Please note that the order of terms in the expression of summation also counts if the terms are distinct.

Take the string $ \large 1{\boxed.} 1\boxed. 1\boxed. 1\boxed. 1$
In the $\;(n-1)\;$ boxes, either put a $+\;$ or a comma.
$1,\;\;1,\;\;1+1+1$ e.g. would represent $1+1+3$
Since you have $2$ choices for each box, # of compositions = $2^{n-1}$
but as you have specified positive integers less than $n$, ans = $2^{n-1} - 1 = 2^4 -1 = 15$
