I am aware of that the number of subsets of a set with n elements is ${2^n}$
However, I tried doing it my own way, by simply listing the number of null, unit, two-element, etc. subsets and adding the total.
So suppose I were to find the number of subsets of a set with 10 elements, then:
Number of null subsets = 1
Number of unit subsets = 10
Number of subsets containing 2 elements= 10 x 9
Number of subsets containing 3 elements = 10 x 9 x 8
Number of subsets containing 4 elements = 10 x 9 x 8 x 7
. . .
Number of subsets containing n elements =10!
Generalising the same for a set containing n elements, the formula looks like
$\sum_ {i=n}^1 \prod_ {j=n}^i j$
Now if the above formula is true, then ideally
$\sum_ {i=n}^1 \prod_ {j=n}^i j$ = ${2^n}$
However, that doesn't seem to be the case. Where did I go wrong, and how can we calculate the number of subsets if we were to go my way?