Carcass wrote:
There are 10 books on a shelf, of which 4 are paperbacks and 6 are hardbacks. How many possible selections of 5 books from the shelf contain at least one paperback and at least one hardback?
A) 75
B) 120
C) 210
D) 246
E) 252
Key observation: It's impossible to have a selection of 5 books, in which none of the books are hardbacks.
In other words, there will always be at least 1 hardback book in a collection of 5 books, which means we just have to deal with having at least one paperbackWell use to formula:
# of ways to obey the restriction = (# of ways to ignore the restriction) - (# of ways to break the restriction) # of ways to ignore the restrictionIn other words, in how many ways can we select 5 books from 10 books?
Since the order in which we select the books does not matter, we can use combinations.
We can select 5 books from 10 books in 10C5 ways
10C5=(10)(9)(8)(7)(6)(5)(4)(3)(2)(1)=(9)(8)(7)(6)(4)(3)(1)=(9)(2)(7)(6)(3)(1)=(3)(2)(7)(6)=252# of ways to break the restrictionIn order to break the restriction, we must have 0 paperbacks in the selection of 5 books,
In other words, we must select 5 books from the 6 hardbacks.
Once again we'll use combinations.
We can select 5 hardbacks from 6 hardbacks in 6C5 ways
6C5=(6)(5)(4)(3)(2)(5)(4)(3)(2)(1)=6So, # of ways to
obey the restriction
=252−6=246Answer: D