Solutions of Question 8 and 9 of Exercise 6.2 of Unit 06: Permutation and Combination. This is unit of Model Textbook of Mathematics for Class XI published by National Book Foundation (NBF) as Federal Textbook Board, Islamabad, Pakistan.
In how many ways can a party of $4$ men and $5$ women be seated at a round table so that no two women are adjacent?
Solution.
If we start seating with women then $5$ women can be seated in (5-1)! arrangements
while $4$ men can be adjusted between seats of women in $41$ ways,
so total possible arrangements $=41 \times 41=576$
How many different signals can be madeWith $2$ blue, $3$ yellow and $4$ green flags using all at a time?
Solution.
$$\text{Total flags} =9$$
$$\text{Repetition of blue }=2$$
$$\text{Repetition of yellow}=3$$
$$\text{Repetition of green}=4$$
\begin{align*}\text{Total signals }&=\dfrac{9!}{2!3!4!}\\
&=\dfrac{362880}{2 \times 6 \times 24}\\
&=\dfrac{362880}{288}=1260\end{align*}