(2) n(B) = 135
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.
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
5470 = 3770 + 3250 - n( F∩W ), so
n( F∩W ) = 3770 + 3250 - 5470 = 1550.
Conclusion: 1550 people use both services.