P&C - Boat question

student4gmat wrote:Question: 8 people are to be seated for rowing a boat in such a way that 4 people will be seated on right side and other 4 on the left side. 3 people are insisting that they will only sit on left side and 2 of them insist to sit only on the right side. Given the condition, how many arrangements are possible?

Let the 8 people be A, B, C, D, E, F, G and H.

Let A, B and C be the people who must sit on the left side.
Let D and E be the people who must sit on the right side.

Number of options for the person joining A, B and C on the left side = 3. (F, G or H.)
Number of ways to arrange the 4 people on the left side = 4! = 24.
The remaining 4 people must sit on the right side.
Number of ways to arrange these 4 people on the right side = 4! = 24.
To combine the options above, we multiply:
3*24*24 = 1728.

GMATGuruNY wrote:

student4gmat wrote:Question: 8 people are to be seated for rowing a boat in such a way that 4 people will be seated on right side and other 4 on the left side. 3 people are insisting that they will only sit on left side and 2 of them insist to sit only on the right side. Given the condition, how many arrangements are possible?

Let the 8 people be A, B, C, D, E, F, G and H.

Let A, B and C be the people who must sit on the left side.
Let D and E be the people who must sit on the right side.

Number of options for the person joining A, B and C on the left side = 3. (F, G or H.)
Number of ways to arrange the 4 people on the left side = 4! = 24.
Number of options for the 2 people joining D and E on the right side = 1. (Of F, G and H, the pair not sitting on the left side.)
Number of ways to arrange the 4 people on the right side = 4! = 24.
To combine the options above, we multiply:
3*24*1*24 = 1728.

Hi GMATGuruNY,
Thank you so much for your reply...I understood the first part that there are 3 ways for people to occupy the 1 space available on the left side. however I did not get the later part (I have highlighted that one in Red colour above).

My question is there there were 3 people left (F, G and H) to occupy the left over seats. After the left is occupied in 3 ways by one of the people we have 2 seats available on the right side and 2 people available (as one is gone to left)so this should be done in 2 ways as per me...I did not get how 1. Please help me understand.

Thanks.

student4gmat wrote: Hi GMATGuruNY,
Thank you so much for your reply...I understood the first part that there are 3 ways for people to occupy the 1 space available on the left side. however I did not get the later part (I have highlighted that one in Red colour above).

In my solution above, I've clarified the reasoning as follows:

Number of options for the person joining A, B and C on the left side = 3. (F, G or H.)
Number of ways to arrange the 4 people on the left side = 4! = 24.
The remaining 4 people must sit on the right side.
Number of ways to arrange the remaining 4 people on the right side = 4! = 24.
To combine the options above, we multiply:
3*24*24 = 1728.