Thread: permutation & combination

hi,can any kind souls please help?thanks loads! =D

ques 1)
three married couples and 10 other people (5 males and 5 females) are randomly selected at a round table. what is the probability that at least one husband will sit next to his wife?

ques 2)
when 12 dice are rolled,what is the probabililty of getting exactly two quadruplets (of identical values)?

Originally Posted by violette

ques 1)
three married couples and 10 other people (5 males and 5 females) are randomly selected at a round table. what is the probability that at least one husband will sit next to his wife?

I will help you with circular arrangements.
Seat one of the non-married people at the table.
Now there are $\displaystyle 15!$ ways to seat the remaining people without any restrictions.
Now we need to find out how many ways to arrange these fifteen people so that at least one married couple are seated together, there are $\displaystyle \sum\limits_{k=1}^3 {\left( { - 1} \right)^{k+1}\binom{3}{k} \left( {2^k } \right)\left[ {\left( {15 - k} \right)!} \right]}$ ways to do that. . That is an inclusion/exclusion solution.

The idea is that we can use the one person already seated at the table as a reference point.

Originally Posted by Plato

I will help you with circular arrangements.
Seat one of the non-married people at the table.
Now there are $\displaystyle 15!$ ways to seat the remaining people without any restrictions.
Now we need to find out how many ways to arrange these fifteen people so that at least one married couple are seated together, there are $\displaystyle \sum\limits_{k=1}^3 {\left( { - 1} \right)^{k+1}\binom{3}{k} \left( {2^k } \right)\left[ {\left( {15 - k} \right)!} \right]}$ ways to do that. . That is an inclusion/exclusion solution.

The idea is that we can use the one person already seated at the table as a reference point.

Hi Plato!Thanks for your help!
But how do I know when I should use the inclusion & exclusion method to solve?

Originally Posted by violette

But how do I know when I should use the inclusion & exclusion method to solve?

This is a rather advanced concept for basic probability students.
I do not know how you are suppose to do this if you have not done a good many of counting problems. To make things more complicated, whoever wrote this question made it a circular arrangement. That is why I gave you more that just hints.