Answer to Question #228117 in Statistics and Probability for Stevie

Question

A pub has 8 different beers on tap. On 4 different days each week they

offer a discount price on one of the beers. The manager is deciding

which beers to discount in a given week, suppose that they only want

to choose the beers, not the specific day each is on special.

(a) If they want to select 4 different beers, how many different choices

are there?

(b) Suppose that they now allow the possibility of selecting the same

beer on multiple days, how many choices are there for selecting

discounts for the week? This means they are choosing some num-

ber of beers, and the number of days each is on special, for a total

of 4. For example one choice might be East End Lager for 2 days

and Adelaide Bitter for 2 days.

(c) If they are still allowing the same beer to be chosen multiple

times, but two of the beers are premium and they want premium

beers to be on special on either 1 or 2 of the 4 days, then how

many choices are there? 

Answer 1

(a) If they want to select 4 different beers

For the first day, they can choose from 8 beers

For the second day, they can choose from 7 beers

For the third day, they can choose from 6 beers

For the fourth day, they can choose from 5 beers

Thus the number of different choices are $=8*7*6*5= 1680$

(b) If the same beer can be chosen for multiple days, then each day, they can choose from 8 beers

Thus the number of different choices are $=8*8*8*8= 4096$

(c) Total number of beers = 8

Number of premium beers = 2, Number of nonpremium beers = 6

If 1 of 4 days, they choose premium to be on special, then the day is chosen in 4 way then, out of 2 premiums, one is chosen to be special in 2 ways, then on the remaining 3 days, each day they can choose from 6 beers

Thus the number of different choices are $=4*2*6*6*6= 432$

If 2 of 4 days, they choose premium to be on special, then the two days is chosen in ⁴C₂= 6 ways

then, each day they can choose from 2 premium beers, then on the remaining 2 days, the day they can choose from 6 beers

Thus the number of different choices are $=6*2*2*6*6= 144$

Therefore, the total number of different choices are 2592