AboutSteve Holleran Expertise I can help with all math questions from basic math to Calculus.
Whether it`s consumer questions, or questions from high school or college students, I have probably dealt with it at some time in my career.
Experience I have taught high school mathematics for the past 33 years in New Jersey. I am now retired in Florida.
Organizations National Council of Teachers of Mathematics
Association of Mathematics Teachers of NJ
Teachers Teaching with Technology
I'm afraid I have a very important national exam tomorrow morning and I would highly value a prompt response. The question is:
Kamil is having a party and needs to seat the seven guests at a round table. He must seat Saad and his class fellow Kasim together, Jamil and Naseem do not want to sit next to each other, but he, himself, must sit with his out of town cousin, Adil. How many seating arrangements are possible?
I've checked up at many places and the only thing I could find about permutations in a circle was that if you have n objects and you want to arrange them in a circle, the total number possible arrangements equals (n-1)! I couldn't find anything on permutations with restrictions.
Regards,
Aftab Khaliq.
Answer Hello Aftab,
I just got your question this morning; I hope my response is not too late to help you.
Counting theory is not exactly my strong point, but I think I've come up with an answer for you. There is a bit of a problem, however, with my understanding of the question. I'm very confused as to exactly how many people are at this dinner party. You first mention Kamil and the seven guests--does this mean 8 people in total? Then as the question goes on, there are only 6 actual names--Kamil, Saad, Kasim, Jamil, Naseem, Adil--does this mean only 6 people total?
Anyway, I think I have an answer for you based on there being the six people you have named. I didn't really use a formula, but thought about the possibilities. One thing I always teach my students in these type of problems is to attend to any special conditions first.
In this problem, there are three sets of special conditions:
1. Saad and Kasim must sit in adjoining seats
2. Jamil and Naseem cannot sit next to each other
3. Kamil and Adil must sit in adjoining seats
I thought of it this way: Let's first seat Saad and Kasim.
Since the table is circular, it doesn't matter where you start, so we can start like this:
Saad(1)
(6) Kasim(2)
(5) (3)
(4)
Now, you cannot put Kamil or Adil in seat #3, because that would mean that 5 and 6 would have to be Jamil and Naseem.
Therefore, seat #3 MUST be either Jamil or Naseem, let's say Jamil. Then #4 and #5 MUST BE Adil and Kamil. It would look like this (one arrangement):
Saad
Naseem Kasim
Adil Jamil
Kamil
To make it easier for me to write, let's list this circular arrangement this way :
Saad--Kasim--Jamil--Kamil--Adil--Naseem
Ok, from this basic setup, there are only several moves:
* interchange Kamil and Adil
(Saad--Kasim--Jamil--Adil--Kamil--Naseem)
* interchange Jamil and Naseem
(Saad--Kasim--Naseem--Adil--Kamil--Jamil)
* switch Kamil and Adil back where they were
(Saad--Kasim--Naseem--Kamil--Adil--Jamil)
These are the only 4 ways for the seating to be set up.
However, there is a total of 8 ways altogether, because you can go through all the ones above, with Saad and Kamil interchanged. This gives another 4, so the total is 8.
It doesn't matter what special condition you start with, because the table is circular. I've worked this out awhile,and I think this is correct.
If I have the number of people incorrect (as I said, I was confused by the wording), perhaps you could let me know, or maybe use this answer to help adjust the result.
Wow!, I didn't anticipate this kind of problem for the weekend! Anyway, I'm sorry if it's not what you need, and I hope it helps, even if only a little.
