Problem 76 - Algebra and Trigonometry

Question

Describe what \(_{n} P_{r}\) represents.

Step-by-Step Solution

Answer

\(_{n} P_{r}\) represents the total number of ways of taking \(r\) elements from a total of \(n\) elements where the order of selection matters. It is calculated using the formula \(_{n} P_{r} = n!/(n-r)!\)

Step 1: Understanding Permutations

A permutation refers to an arrangement of items where the order is important. So, if we have \(n\) items, how many ways can we arrange \(r\) of these items? This is what \(_{n} P_{r}\) represents.

Step 2: Explaining the Permutation Formula \(_{n} P_{r}\)

\(_{n} P_{r}\) is calculated using the formula \(_{n} P_{r} = n!/(n-r)!\). Here, \(n!\) denotes the factorial, meaning the product of all positive integers up to \(n\). For example, 5! = \(5 × 4 × 3 × 2 × 1\). The symbol \((n - r)!\) calculates the factorial of \((n-r)\).

Step 3: Interpreting the Formula

\(_{n} P_{r}\) represents the total number of ways of taking \(r\) elements from a total of \(n\) elements where the order of selection matters. For example, in a race, the number of ways in which the top 3 places can be decided amongst 10 participants would be \(_{10} P_{3}\).