Question

How many subsets of a set with ten elements

a) have fewer than five elements?

b) have more than seven elements?

c) have an odd number of elements?

Short answer

a) There are 386 subsets fewer than five elements.

b) There are 56 subsets more than seven elements.

c) There are 512 subsets have an odd number of elements.

Step-by-step solution

Introduction

Subsets are a component of the Sets notion, which is a mathematical concept.

Explanation

(a)

Every set with \({\rm{n}}\) items has precisely \(\left( {\begin{array}{*{20}{l}}{\rm{n}}\\{\rm{k}}\end{array}} \right)\) number of \({\rm{k}}\) element subsets. The solutions are self-explanatory if you keep this in mind:

The number of subsets having less than five members is large.

\(\begin{array}{l}{\rm{ = }}\sum\limits_{{\rm{i = 0}}}^{\rm{4}} {\left( {\begin{array}{*{20}{c}}{{\rm{10}}}\\{\rm{i}}\end{array}} \right)} \\{\rm{ = 1 + 10 + 45 + 120 + 210}}\\{\rm{ = 386}}{\rm{.}}\end{array}\)

Hence, there are 386 subsets fewer than five elements.

Explanation

b)

The number of subgroups with seven or more elements

\(\begin{array}{c}{\rm{ = }}\sum\limits_{{\rm{i = 8}}}^{{\rm{10}}} {\left( {\begin{array}{*{20}{c}}{{\rm{10}}}\\{\rm{i}}\end{array}} \right)} \\{\rm{ = 45 + 10 + 1}}\\{\rm{ = 56}}{\rm{.}}\end{array}\)

Therefore, there are 56 subsets more than seven elements.

Explanation

c)

The number of subsets with an odd number of elements is equal to half of the total number of subsets (implication of binomial identity)\({\rm{ = }}\frac{{\rm{1}}}{{\rm{2}}}{\rm{.}}{{\rm{2}}^{{\rm{10}}}}{\rm{ = 512}}\).

Thus, there are 512 subsets have an odd number of elements.