Permutations and Combinations

Question

1) A man and two women are to sit in a row of five empty chairs.Calculate the number of ways they can be seated if
(i) the two women must sit next to each other (Ans 24)
(ii) all 3 people must be sit next to each other (Ans 18)

Answer 1

Judging by your answers an order of people matters.

 

(i) Let us consider 2 women sitting together as one entity. So at first stage we will consider permutation of 4 entities: one man, two women and 2 empty chairs in one line. (We will take into consideration permutation of women at second stage)

There are 4! permutations. Now an empty chair order does not matter, thus we should divide it by 2!, which is the number of all permutations of 2 chairs.

4!/2!=12.

Second stage: for each sitting we can permute women and get a new arrangement. Therefore the complete number is 12*2!=24.

 

(ii) Now we consider 3 entities to permute, it will be 3!=6. Order of 2 chairs does not matter: 6:2!=3.

For each computed permutation we can permute 3 people, and the number of all permutations is 3!=6.

Hence the answer is 3*6=18.