Question

How many subsets with more than two elements does a set with 100elements have?

Short answer

There are $2^{100}-5051$subsets with more than two elements does a set with 100 elements have.

Step-by-step solution

Given data

A set with 100 elements.

Concept of Combination

A combination is a selection of items from a set that has distinct members.

Formula:

${}_{\mathbf{n}}{\mathbf{C}}_{\mathbf{r}}\mathbf{=}\frac{\mathbf{n}\mathbf{!}}{\mathbf{r}\mathbf{!}\mathbf{(}\mathbf{n}\mathbf{-}\mathbf{r}\mathbf{)}\mathbf{!}}$

Calculation for the subsets

There are $2^{100}$total number of subsets of a set with 100 elements. The number of subsets with 0 elements is 1, the number subsets with 1 element is given as:

$\begin{array}{r}C(100,1)=\frac{100!}{(1\times 991!)}\\ =100\end{array}$

The number of subset with 2 elements is given as:

$\begin{array}{r}C(100,2)=\frac{100!}{(2!\times 981!)}\\ =4950\end{array}$

Thus, the number of subsets with more than two elements is

$2^{100}-1-100-4950=2^{100}-5051$

Thus, there are $2^{100}-5051$subsets with more than two elements does a set with 100 elements have.