Question

How many subsets with an odd number of elements does a set withelements have?

Short answer

There are 512 subsets with an odd number of elements does a set with 10 elements have.

Step-by-step solution

Given data

An odd number of elements.

 Step 2: 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 of subsets with an odd number of elements

Let take 1 to 10 number of elements as it contain 5 odd numbers 1,3,5,7 and 9.

Hence, possible subsets are given as:

$\begin{array}{r}\mathrm{C}(10,1)+\mathrm{C}(10,3)+\mathrm{C}(10,3)+\mathrm{C}(10,3)+\mathrm{C}(10,9)=\left(2^{10}\right)/2\\ =2^9\\ =512\end{array}$

Thus, there are 512 subsets with an odd number of elements does a set with 10 elements have.