Chapter 13.9, Problem 34HP

To determine

To find: Given statement is true or false, is false provide a counterexample.

Answer to Problem 34HP

Hence, the statement that the number of combinations of items is always greater than $1$ is false.

Explanation of Solution

Given information:

The number of combinations of items is always greater than $1$ .

It is given that The number of combinations of items is always greater than $1$ .

Given statement is false and the counterexample is given as below:

Assume there are $n$ types of butter toppings available for a pizza base.

Of a person orders a pizza without any topping.

Then, clearly this is an example of combinations as there is no arrangement required.

Then selecting a pizza base such that it does not have any butter topping is calculated as $C\left(n,0\right)$ .

  $\begin{array}{l}C\left(n,0\right)=\frac{P\left(n,0\right)}{Totalnumberofcases}\\ =\frac{n}{n}\\ =1\end{array}$

That is , the no. of combinations of items here is equal to $1$ .