Question

The number 3 can be written as 3,2+1,1+2 or 1+1+1 in four ways. In how many ways the number n can written

Answer 1

In infinitive ways because it's infinite

Answer 2

This can be calculate by using Hardy- Ramanujan Partition Theory

This is a case of using the partitions.

Partitions are the forms of any number in which you can represent the number as sum of numbers less than it.

The Ramanujan Formula for finding the partitions of a number is given as:

$\sum _{n=0}^{\infty }q(n)x^{n}=\prod _{k=1}^{\infty }(1+x^{k})=\prod _{k=1}^{\infty }{\frac {1}{1-x^{2k-1}}}.$

This was developed by ramanujan and GH hardy in 1918.