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Q23E

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Discrete Mathematics and its Applications
Found in: Page 414
Discrete Mathematics and its Applications

Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095

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Short Answer

How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other? (Hint: First position the men and then consider possible positions for the women.)

The total number of possible arrangements is \(609,638,400\).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :
TABLE OF CONTENTS

Step 1: Given data

Number of men \( = 8\) and number of women \( = 5\).

Step 2: Concept of Permutation

The word "permutation" refers to the act or process of changing the linear order of an ordered set.

Formula:

\(_n{P_r} = \frac{{n!}}{{(n - r)!}}\)

Step 3: Calculation to find the position of men

First, consider the position of men.

Find the possible ways to arrange men in a row.

\(\begin{array}{l}P(8,8) = \frac{{8!}}{{(8 - 8)!}}\\P(8,8) = \frac{{8!}}{{0!}}\\P(8,8) = 40,320\end{array}\)

It is given that no two women stand next to each other.

The situation becomes:

\(O\;M\;O\;M\;O{\rm{ }}M\;O\;M\;O\;M\;O{\rm{ }}M\;O\;M\;O\;M\;O\)

Step 4: Calculation to find the number of possible arrangement

There are \(9\) places for women. We can arrange \(5\) women in these 9 places.

Now, find the ways to place women:

\(\begin{array}{l}P(9,5) = \frac{{9!}}{{(9 - 5)!}}\\P(9,5) = \frac{{9!}}{{4!}}\\P(9,5) = 15,120\end{array}\)

Find the total number of possible arrangements:

Number of possible arrangements \(\; = 15,120 \times 40,320\)

Number of possible arrangements \( = 609,638,400\)

Hence, the total number of possible arrangements is \(609,638,400\).

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