Question

A professor writes \({\rm{20}}\) multiple-choice questions, each with the possible answer a, b, c, or d, for a discrete mathematics test. If the number of questions with a, b, c, and d as their answer is \({\rm{8, 3, 4, and 5,}}\)respectively, how many different answer keys are possible, if the questions can be placed in any order?

Short answer

The required answer keys are\({\rm{3,491,888,400}}\).

Step-by-step solution

Introduction

It is possible to distribute\({\rm{n}}\)identifiable items into k distinguishable boxes in such a way that\({{\rm{n}}_{\rm{i}}}\)objects are put in box\({\rm{i(i = 1,2,3,4,5)}}\)in ways\(\frac{{{\rm{n!}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{!}}{{\rm{n}}_{\rm{2}}}{\rm{! \ldots }}{{\rm{n}}_{\rm{k}}}{\rm{!}}}}\).

Explanation

There are\({\rm{20}}\)multiple-choice questions with \({\rm{8}}\) times \({\rm{a}}\), 3 times \({\rm{b}}\), \({\rm{4}}\) times \({\rm{c}}\), and \({\rm{5}}\) times \({\rm{c}}\) answers.

\(\begin{array}{l}{\rm{n = 20}}\\{\rm{k = 4}}\\{{\rm{n}}_{\rm{1}}}{\rm{ = 8}}\\{{\rm{n}}_{\rm{2}}}{\rm{ = 3}}\\{{\rm{n}}_{\rm{3}}}{\rm{ = 4}}\\{{\rm{n}}_{\rm{4}}}{\rm{ = 5}}\end{array}\)

It is possible to distribute\({\rm{n}}\)identifiable items into k distinguishable boxes in such a way that \({{\rm{n}}_{\rm{i}}}\) objects are put in box \({\rm{i(i = 1,2,3,4,5)}}\) in ways:

\(\frac{{{\rm{n!}}}}{{{{\rm{n}}_{\rm{1}}}{\rm{!}}{{\rm{n}}_{\rm{2}}}{\rm{! \ldots }}{{\rm{n}}_{\rm{k}}}{\rm{!}}}}\).

\(\frac{{{\rm{20!}}}}{{{\rm{8!3!4!5!}}}}{\rm{ = 3,491,888,400}}\).

Therefore, the required answer keys are\({\rm{3,491,888,400}}\).