In how many ways can 8 people be seated in a row if a. There are no restrictions
ID: 2946090 • Letter: I
Question
In how many ways can 8 people be seated in a row if a. There are no restrictions on the seating arrangement? 1. Sol b. Person A and B must sit next to each other? Sol There are 4 men and 4 women and the no 2 men or no 2 women can sit next to each other Sol c. d. There are 5 men and they must sit next to each other? Sol e. There are four married couples and each couple must sit together? Sol A student has to sell 2 books from a collection of 6 math books, 7 science books, and 4 economics books. How many choice are possible if a. Both books are to be on the same subiect? 2. Sol b. The books are to be on different subject? Sol 3. An urn contains 4 red balls and 3 green balls. Draw three balls, what is the probability that two balls are red and another ball is green.Explanation / Answer
a)
If there are no restriction then we count all possible permutations which are 8! = 40,320
b) If A and B sit together then they constitute a "block" which can be permutation along with the remaining 6 people
in 7! different ways. Inside this block there are 2!=2 possibilities of permuting A and B so due to the basic counting
principle we get a total number of 2*7! = 10,080 ways
c) The restrictions impose having persons of opposite sex next to each other. We shall apply here the generalized principle of counting in the following way:
on the first position we can have any of the 8 persons, then for one choice of the person for the first position we shall have 4 possible choices for second position amoung the people of opposite sex, then for any particular choice of the first two positions we have 3 choices for the third position amoung the remaining people of the same sex with the person on the first position (and consequently of different sex with the one on the second position) and so on....
So multiplying all these numbers we get 8*4*3*3*2*2*1*1 = 1152 ways of arranging the people
d) the men can be considered again a "block" and permuted along with the remaining 3 women in 4! ways and as we can also permuted the men within the block in 5! ways then the total amounts to 5!*4! = 2880 ways
e) within each couple there are 2!=2 possibilities of siting the people so a total of 2*2*2*2 = 16 ways. As we can also permute the couples amoung themselves in 4! ways then we have 16*4! = 384 ways of arranging the people