A stereo store is offering a special price on a complete set ofcomponents (receiver, compact disc player, speakers, cassette dec

Question

2 months ago

10

A stereo store is offering a special price on a complete set ofcomponents (receiver, compact disc player, speakers, cassette dec

k)ie one of each. A purchaser is offered a choice ofmanufacturer for each component:
Receiver: Kenwood, Onkyo, Pioneer, Sony, Sherwood
CD Player: Onkyo, Pioneer, Sony, Technics
Speakers: Boston, Infinity, Polk
Cassette Deck: Onkyo, Sony, Teac, Technics

A switch board display in the store allows a customer to hooktogether any selection of components (consisting of one of eachtype). Use the product rules to answer the following questions.
a. In how many ways can one component of each type beselected?
b. In how many ways can components be selected if boththe receiver and the cd player are to be sony?
c. In how many ways can components be selected if noneis to be sony?
d. In how many ways can a selection be made if at leastone sony component is to be included?
e. If someone flips the switches on the selection in acompletely random fashion, what is the probability that the systemselected contains at least one sony component? Exactly onesony component?

Mathematics

1 answer:

zzz [12.3K] · Answer 1 · 2026-03-16T11:17:01+03:00

Response:

Detailed explanation:

(a)

There are 5 choices for receivers.

There are 4 choices for CD players.

There are 3 choices for speakers.

There are 4 choices for cassettes.

You can choose one receiver from 5 options in $5C_1$ ways.

You can choose one CD player from 4 options in $4C_1$ ways.

You can choose one speaker from 3 options in $3C_1$ ways.

You can choose one cassette from 4 options in $4C_1$ ways.

Determine the number of combinations for selecting one component of each category.

According to the multiplication rule, the total number of ways to choose one component from each type is

The total number of combinations for selecting one component from each type is

$=5C_1*4C_1*3C_1*4C_1\\\\=5*4*3*4\\\\=240$

Part a

Thus, the total number of ways to select one from each type is 240.

(b)

There is 1 Sony receiver available.

There is 1 Sony CD player available.

There are 3 speakers available.

There are 4 cassettes available.

You can select one Sony receiver from 1 available in ways.

You can select one Sony CD player from 1 available in ways.

You can select one speaker from 3 options in ways.

You can choose one cassette from 4 options in $4C_1$ ways.

Calculate the number of ways to select components if the receiver and CD player must both be Sony.

Applying the multiplication rule, the total number of ways to select components when both the receiver and CD player are Sony is

Count of combinations for selecting one component of each type

$=1C_1*1C_1*3C_1*4C_1\\\\=1*1*3*4\\\\=12$

Consequently, the total number of combinations for selecting components when both the receiver and CD player are Sony is 12.

(c)

The count of receivers excluding Sony is 4.

The count of CD players excluding Sony is 3.

The count of speakers excluding Sony is 3.

The count of cassettes excluding Sony is 3.

You can select one receiver from the 4 available in 4C_1 ways.

You can choose one CD player from the 3 available in 3C_1 ways.

You can select one speaker from the 3 available in 3C_1 ways.

You can select one cassette from the 3 available in 3C_1 ways.

Determine the total number of ways to select components when none are to be Sony.

Utilizing the multiplication rule, the number of ways to select components without including Sony is

$=4C_1*3C_1*3C_1*3C_1\\\\=108$

Thus, the total number of selections for non-Sony components is 108.

(d)

The number of combinations for making a selection that includes at least one Sony component is,

= Total selections possible - Selections without any Sony components

= 240-108

= 132

Therefore, the total number of ways to make a selection including at least one Sony component is 132.

(e)

In the event that someone switches the selection randomly, the probability that the chosen system includes at least one Sony component is,

$= \text {Total possible selections with at least one Sony} /\text {Total possible selections}$

= 132 / 240

= 0.55

The probability that the selected system contains exactly one Sony component is,

$= \text {Total possible selections with exactly one Sony} /\text {Total possible selections}$ $\frac{1C_1*3C_1*3C_1*3C_1+4C_11C_13C_13C_1+4C_13C_13C_13C_1}{240} \\\\=\frac{99}{240} \\\\=0.4125$

Therefore, if someone alters the selection switches randomly, the probability that the system includes at least one Sony component is 0.55.

If someone randomly flips the switches on the selection, the probability that the chosen system has exactly one Sony component is 0.4125.