Suppose that a teacher is going to assign a book report to h

Suppose that a teacher is going to assign a book report to her class of 31 students. Each student must select one book from an approved reading list and once a book is selected by a student, no other student may select the same book. The reading list consists of a total of 40 books of which 27 are considered classic fiction and the other 13 are considered modern fiction. Assuming that the order in which the books are selected doesn’t matter:

a) In how many ways can the books be selected so that all of the modern fiction books are picked?

b) In how many ways can the books be selected so that there is at least one modern fiction book that is not picked and at least one classic fiction book is not picked?

Solution

There are alltogether 40 books(call units).We classify them as A=Classic fiction,B=Modern fiction.

As the no. of students is 31,we are basically selecting thereby 31 books(sample size drawn) without replacement according to that \"a book once selected by one can not be selected by others\".

(a) All of the B category units are sampled => the remaining (31-13 =) 18 units are to be selected from the A category.Which is possible by (²⁷C₁₈)*(¹³C₁₃) = (4686825)*(1) ways.

(b) The event { there is at least one modern fiction book that is no picked and there is at least one classic fiction book that is not picked } can occur in the following combinations from the two categories of book-->

{ 26 A units + (31-26=) 5 B units}U{ 25 A units + (31-25 =) 6 B units}U{ 24 A units + (31-24=) 7 B units}U...U{19 A units + 12 B units}.

The no. of ways is (²⁷C₂₆)*(¹³C₅) + (²⁷C₂₅)*(¹³C₆) + (²⁷C₂₄)*(¹³C₇) +....+ (²⁷C₁₉)*(¹³C₁₂) [since all the events are disjoint i.e, can not occur simultaneously, hence additive]