Question

In how many ways can 4 ladies and 5
gentlemen be seated in a row so that no two
ladies sit together?
​

Answer 1

Step-by-step explanation:

fjkjzfgjnnkllhxaz bmkp

Answer 2

Answer: Arrange all of the men in a row

We can arrange k unique objects in k! ways.

Since there are 5 men, we can arrange them in 5! (120 ways)

IMPORTANT: Now place an empty chair on either side of each man as follows:

_M_M_M_M_M_

Note: This prevents the women from sitting together because there is now a man separating each of 6 empty chairs.

Stage 2: Seat a woman

There are 6 seats, so we can complete this stage in 6 ways

Stage 3: Seat another woman

There are 5 seats remaining, so we can complete this stage in 5 ways

Stage 4: Seat another woman

There are 4 seats remaining, so we can complete this stage in 4 ways

Stage 5: Seat the last woman

There are 3 seats remaining, so we can complete this stage in 3 ways

By the Fundamental Counting Principle (FCP), we can complete all 5 stages (and thus seat all 9 people) in (120)(6)(5)(4)(3) ways ([spoiler]= 43200, ways[/spoiler])