1. ## permutation & combination

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)?

2. 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?
Seat one of the non-married people at the table.
Now there are $15!$ 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 $\sum\limits_{k=1}^3 {\left( { - 1} \right)^{k+1}\binom{3}{k} \left( {2^k } \right)\left[ {\left( {15 - k} \right)!} \right]}$ 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.

3. Originally Posted by Plato
Seat one of the non-married people at the table.
Now there are $15!$ 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 $\sum\limits_{k=1}^3 {\left( { - 1} \right)^{k+1}\binom{3}{k} \left( {2^k } \right)\left[ {\left( {15 - k} \right)!} \right]}$ 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.