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Expert-verified Found in: Page 414 ### Discrete Mathematics and its Applications

Book edition 7th
Author(s) Kenneth H. Rosen
Pages 808 pages
ISBN 9780073383095 # 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 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 $$609,638,400$$. ### Want to see more solutions like these? 