rajatvmittal wrote:A photographer will arrange 6 people for photograph by placing them in two rows of three. how many arrangements are possible?

A photographer will arrange 6 people for photograph by placing them in a row.how many arrangements are possible?

The answers to these two questions will be the same.

For the first question, we have 6 spaces to fill. These spaces can be numbered as follows:
1 2 3
4 5 6

For the second question, we have 6 spaces to fill. These spaces can be numbered as follows:
1 2 3 4 5 6


We'll take the task of seating the 6 people and break it into stages.

Stage 1: Choose someone to sit in chair #1.
There are 6 people, so this stage can be accomplished in 6 ways.

Stage 2: Choose someone to sit in chair #2.
There are now 5 unseated people remaining, so this stage can be accomplished in 5 ways.

Stage 3: Choose someone to sit in chair #3.
There are now 4 unseated people remaining, so this stage can be accomplished in 4 ways.

Stage 4: Choose someone to sit in chair #4.
There are now 3 unseated people remaining, so this stage can be accomplished in 3 ways.

Stage 5: Choose someone to sit in chair #5.
There are now 2 unseated people remaining, so this stage can be accomplished in 2 ways.

Stage 6: Choose someone to sit in chair #6.
There is now 1 unseated person remaining, so this stage can be accomplished in 1 ways.

By the Fundamental Counting Principle (FCP) we can complete all 6 stages (and thus seat all 6 people) in (6)(5)(4)(3)(2)(1) ways ([spoiler]= 6! ways = 720 ways[/spoiler])

Cheers,
Brent

Aside: For more information about the FCP, we have a free video on the subject: https://www.gmatprepnow.com/module/gmat-counting?id=775