combinations & permutations

[carebear]

If there are 3 strangers that choose a hotel in town with 7 hotels....

If one of the hotels has room for only 2 more people , in how many ways could the three strangers find accommodations?

Is the correct answer 342 ....for all possible combinations minus 1 (343-1) OR

294......7 x 7 x 6?

 

[soroban]

Hello, carebear!

I get 342 . . .

There are 3 strangers that choose hotels in town with 7 hotels.

If one of the hotels has room for only 2 more people,
in how many ways could the three strangers find accommodations?

There is a booked-up hotel which has only 2 vacancies; call the hotel \(\displaystyle H.\)
The other six hotels can accomodate at least three more guests.

There are 3 cases to consider.

(1) Two of the strangers go to \(\displaystyle H.\)
. . .There are: \(\displaystyle _3C_2 = 3\) choices for which two go to \(\displaystyle H.\)
. . .The third stranger has a choice of the other 6 hotels.
. . .There are: .\(\displaystyle 3\cdot6 \:=\:18\) ways.

(2) One stranger goes to \(\displaystyle H.\)
. . .There are \(\displaystyle 3\) choices for the one who goes to \(\displaystyle H.\)
. . .There two have 6 choices each for their hotels: .\(\displaystyle 6^2\) choices.
. . .There are: .\(\displaystyle 3\cdot 6^2 \:=\:108\) ways.

(3) No stranger goes to \(\displaystyle H.\)
. . .Then each stranger has a choice of the other 6 hotels.
. . .There are: .\(\displaystyle 6^3 = 216\) ways.


\(\displaystyle \text{Therefore, there are: }\;18 + 108 + 216 \:=\:342\text{ possible accomodations.}\)