<s>[INST] find the total number of possible way to reach to a particular sum suppose you have given a sum like : 5.
we have to find the total number of possible way to reach to 5.
for example
1 + 1 + 1 + 1 + 1 = 5
2 + 1 + 1 + 1     = 5
1 + 2 + 1 + 1     = 5
1 + 1 + 2 + 1     = 5
1 + 1 + 1 + 2     = 5
2 + 2 + 1         = 5
2 + 1 + 2         = 5
1 + 2 + 2         = 5
1 + 1 + 3         = 5
1 + 3 + 1         = 5
3 + 1 + 1         = 5
2 + 3             = 5
3 + 2             = 5
1 + 4             = 5
4 + 1             = 5
5                 = 5
above is the total number of way to reach to a particular sum, which is nothing but : 16
 [/INST] By Stars and Bars the number of ordered $k-$tuples of positive integers that sum to $n$ is $\binom {n-1}{k-1}$.  It follows that your answer is $$\sum_{k=1}^n\binom {n-1}{k-1}=\sum_{k=0}^{n-1}\binom {n-1}k=2^{n-1}$$
Direct Proof:  List $n$ $*'s$ with blanks between them.  There are $n-1$ blanks.  your decompositions are entirely determined by an arbitrary subset of those $n-1$ blanks.
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