Section 6-1

(2) n(B) = 135

(4) n(A') = 185
(6) n( A ∩ B ) = 40
(8) n( A ∩ B' ) = 75
(10) n( A' ∩ B' ) = 90
(12) n( (A ∪ B)' ) = 90
(28)
There are 3 operations here: choose bread, choose meat, choose lettuce or sprouts. The number of outcomes of these operations is 3, 5, and 2, respectively. Thus, by the multiplication principle, the total number of outcomes is (3)(5)(2) = 30.

(30)
How many 5-letter code words can be made from the letters ABCDEFG.

(a) if no letter is repeated.
By multiplication principle: (7)(6)(5)(4)(3) = 2520.

(b) if letters can be repeated.
By multiplication principle: (7)(7)(7)(7)(7) = 16,807.

(c) if adjacent letters must be different.
By multiplication principle: (7)(6)(6)(6)(6) = 9,072.

(48)
There are 10,000 customers.
3,770 use call forwarding.
3,250 use call waiting.
4,530 use neither service.
How many customers use both services?

Let F be the set of people who use call forwarding. Let W be the set of people who use call waiting. Let U be the set of all 10,000 customers. Now, F∪W is the set of people who use one or both of the services, so it's the complement of the set of people who use neither service. Therefore n( F∪W ) = 10,000 - 4,530 = 5470.

We want to find out how many people use both services, in other words how many elements are in the set F∩W.

By the addition principle, n( F∪W ) = n( F ) + n( W ) - n( F∩W ), which turns into
5470 = 3770 + 3250 - n( F∩W ), so
n( F∩W ) = 3770 + 3250 - 5470 = 1550.

Conclusion: 1550 people use both services.