A stereo store is offering a special price on a complete set

A stereo store is offering a special price on a complete set of components (receiver, compact disc player, speakers, turntable). A purchaser is offered a choice of manufacturer for each component:

A switchboard display in the store allows a customer to hook together any selection of components (consisting of one of each type). Use the product rules to answer the following questions:

(a) In how many ways can one component of each type be selected?
1 ways

(b) In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?
2 ways

(c) In how many ways can components be selected if none is to be Sony?
3 ways

(d) In how many ways can a selection be made if at least one Sony component is to be included?
4 ways

(e) If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component? Exactly one Sony component? (Round your answer to three decimal places.)

exactly one Sony component

Solution

(a) In how many ways can one component of each type be selected?

To find the number of ways available we will multiply the possible variations which are avaiilable for each case.

Number of ways = 4*4*3*4 = 192

(b) In how many ways can components be selected if both the receiver and the compact disc player are to be Sony?

We have now fixed the variations in reciever and compact disc player, so here we will only multiply the variations available in speakers and turntable.

Number of ways = 1*1*3*4 = 12

(c) In how many ways can components be selected if none is to be Sony?

Now we will exclude all the sony items from variations.

Number of ways = 3*3*3*3 = 81

(d) In how many ways can a selection be made if at least one Sony component is to be included?

This can be found by reducing the number of ways where no sony component is included from the total number of ways.

Number of ways = 192 - 81 = 111

(e)

If someone flips switches on the selection in a completely random fashion, what is the probability that the system selected contains at least one Sony component?

P = (number of ways to select at least one sony component)/(total number of ways) = 111/192 = 0.578

Exactly one Sony component?

P = (number of ways to select exactly one sony component)/total number of ways = (3*3*3+3*3*3+3*3*3)/192 = 0.422