Question

Explain why \(_{3} P_{5}\) gives an error message when evaluated on a calculator.

Short answer

\( _{3} P_{5} \) gives an error because \( (-2)! \) is undefined.

Step-by-step solution

Understand the Notation

The notation \( _{3} P_{5} \) represents the number of permutations of 5 items taken from a set of 3 items. It is calculated using the permutation formula \( _{n} P_{r} = \frac{n!}{(n-r)!} \).

Identify Values of n and r

In this notation, \( n = 3 \) and \( r = 5 \). The formula then becomes \(_{3} P_{5} = \frac{3!}{(3-5)!} \).

Calculate the Factorials

Compute the factorials involved: \( 3! = 6 \) and \( (3-5)! = (-2)! \).

Recognize the Problem

Notice that factorials are only defined for non-negative integers. The term \( (-2)! \) is not defined, which causes the formula to be invalid.

Explain the Calculator Error

When evaluating \( _{3} P_{5} \) on a calculator, it attempts to compute \( (-2)! \), which is undefined. This results in an error message.

Key concepts

Permutation Notation

Permutation notation is used to represent the number of ways to arrange a subset of items from a larger set. For example, the notation \(_{n} P_{r}\) indicates the number of permutations of \(r\) items taken from a set of \(n\) items. To put it simply, permutations are about arranging items in a specific order. In \(_{n} P_{r}\), \(n\) is the total number of items you can choose from, and \(r\) is the number of items you want to arrange.

The formula to calculate permutations is defined as: \[ _{n} P_{r} = \frac{n!}{(n-r)!} \] where \(!\) denotes factorial, a mathematical function we'll explain next. This notation helps in understanding how many different ways you can order \(r\) items out of \(n\).

Factorials

A factorial is a product of all the positive integers up to a given number. The symbol for factorial is \(!\). For instance, \(3!\) means \(3 \times 2 \times 1 = 6\). It's a simple way to calculate the number of possible permutations for a given number.

Here are some basic examples of factorials:

• \(0! = 1\) - By definition
• \(1! = 1\)
• \(2! = 2 \times 1 = 2\)
• \(3! = 3 \times 2 \times 1 = 6\)

When dealing with permutations, calculating factorials is essential. Factorials are defined only for non-negative integers. When you try to calculate a factorial for a negative number, it is undefined, leading to issues, such as errors on calculators.

Calculator Error

Calculators are programmed to follow the rules of mathematical operations. When you input an operation that involves an undefined mathematical concept, the calculator displays an error.

In the case of \(_{3} P_{5} = \frac{3!}{(3-5)!}\), the calculation involves finding \((-2)!\), which is not defined.

Since factorials for negative numbers do not exist, this leads to an error on the calculator. When the calculator tries to compute this, it cannot complete the operation, resulting in an error message.

Always remember to check the mathematical validity of your operations. If you encounter an error, it often means that you've tried to perform an operation that math rules do not allow.

Non-negative Integers

Non-negative integers are all the whole numbers starting from zero and extending to positive infinity. They include numbers like \(0, 1, 2, 3, ...\). Factorials are only defined for these integers.

In the context of permutations, both \(n\) and \(r\) should be non-negative integers. This ensures that the factorial calculations involved in permutations are valid.

If either \(n\) or \(r\) is negative, it can lead to invalid expressions, like factorials of negative numbers, which are undefined. This is why in the exercise \(_{3} P_{5}\), the negative factorial causes an error. Always ensure that values you use are within the valid range of non-negative integers to avoid such issues.