Paige Turner loves finite mathematics. She has 5 books about probability and 4 books about matrices. Her friend Anita Tudor wants to borrow four of these books.
How many ways can Paige choose the books to lend Anita if:
a) there are no restrictions?
9 books CHOOSE 4 = 9C4 = (9∙8∙7∙6)/(4∙3∙2∙1) = 3024/24 = 126 ways
b) there are two books on each topic (i.e. two matrix books and two probability books)?
(5 probability books CHOOSE 2) AND (4 matrix books CHOOSE 2) = (5C2)∙(4C2) =
[(5∙4)/(2∙1)]∙[(4∙3)/(2∙1)] = [20/2]∙[12/2] = 10∙6 = 60 ways
c) there is at least one book on each topic?
Whenever we see the words "at least", we proceed in two steps:
1. Find the number of ways to select ANY 4 books.
2. Find the total number of UNWANTED selections of 4 books.
3. Subtract the second number from the first.
1. This was the answer to part (a) with no restrictions, or 126 ways
2. There are two types of UNWANTED selections, so we will have to calculate
both types of UNWANTED selections. The 2 types of unwanted selections
are:
A. When all 4 books are probability books and there are no matrix books.
B. When all 4 books are matrix books and there are no probability books.
A. That's 5 probability books CHOOSE 4 = 5C4 = (5∙4∙3∙2)/(4∙3∙2∙1) =
120/24 = 5 ways
B. That's 4 matrix books CHOOSE 4 = 4C4 = (4∙3∙2∙1)/(4∙3∙2∙1) =
24/24 = 1 way
So the total number of UNWANTED ways is 5+1 or 6
3. Subtract 6 from 126 and get 126-6 = 120.
Edwin