There are n couples invited to a banquet Suppose there are s

There are n couples invited to a banquet. Suppose there are some number of speeches given during the banquet, so that at most one member of each couple gives a talk. How many different sets of speakers could the evening have?

Solution

n couple would mean we have n men and n women. Lets assume that this fact holds( universal fact)

Now, at most 1 member of each couple gives a talk. This would mean that we will have at most 1 from each couple giving a sppech during banquet.

We can have no people talking , 1 person talking from the whole n couple group, 2 people,................ or each of n couples pair giving a speech =

Hence, the difference sets one can have : nC0 +nC1 +nC2 + ....... n Cn = 2^n

Please note that if we neccesarily got to have speech then the first term( nC0 = 1) will yeild to 0 and therefore come to the RHS. Hence, nC1 +nC2 + ....... n Cn = 2^n -1

Therefore, 2^n ways when atmost 1 from each couple pair, or 2^n -1 when we need atleast 1 speech from the group, will leave it to you to decide on this one.