Question

Mary has on her bookshelf 5 novels, 4 biographies, and 8 textbooks.

Mary wants to take a fiction and a non-fiction book with her on a short trip.

(a) How many different ways can she do this?

(b) Mary thinks a little and then decides that she wants instead to take a novel and a biography. How many different ways can she do this?

(c) On a longer trip Mary decides to take three novels and four non-fiction books with at least one of the non-fiction books a biography. How many ways are there to make such a selection?

Answer 1

Assumption: All novels are fictional and rest of the books (biographies and textbooks) are non-fictional.

formula: choose k product from n choices = C(n,k) = ⁿC_k =(n!) / (k!*(n-k)!)

(a)

a fiction and a non-fiction book

total options available for fiction book = 5

total options available for non fiction book = 4+8 = 12

so total ways to do = ⁵C₁ * ¹²C₁ = 5 * 12 = 60

so 60 is the answer.

(b)

a novel and a biography

total options available for novels = 5

total options available for biographies = 4

so total ways to do = ⁵C₁ * ⁴C₁ = 5 * 4 = 20

so 20 is the answer.

(c)

three novels and four non-fiction books with at least one of the non-fiction books a biography

total ways to choose three novels =  ⁵C₃ = 10

ways to choose 4 non fictions books with atleast one biography

= ways to choose 4 non fictions (any) - ways to choose 4 non fictions (with no biography)

=   ¹²C₄ -   ⁸C₄

= 425

so total ways to do = ways to choose 3 novels * ways to choose 4 non finction books with atleast one biography

= 10*425 = 4250

so 4250 is the answer.