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A professor writes 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 8,3,4,and5,respectively, how many different answer keys are possible, if the questions can be placed in any order?

Short Answer

Expert verified

The required answer keys are3,491,888,400.

Step by step solution

01

Introduction

It is possible to distributenidentifiable items into k distinguishable boxes in such a way thatniobjects are put in boxi(i=1,2,3,4,5)in waysn!n1!n2!โ€ฆnk!.

02

Explanation

There are20multiple-choice questions with 8 times a, 3 times b, 4 times c, and 5 times c answers.

n=20k=4n1=8n2=3n3=4n4=5

It is possible to distributenidentifiable items into k distinguishable boxes in such a way that ni objects are put in box i(i=1,2,3,4,5) in ways:

n!n1!n2!โ€ฆnk!.

20!8!3!4!5!=3,491,888,400.

Therefore, the required answer keys are3,491,888,400.

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