1. ## Combinatorics

hi

i don't know where else to post this but i need some help on a textbook question i was given about combinatorics.

there is a game that involves a murderer, a mansion, and the weapon used to commit the murder. there are 5 characters in all, 7 rooms in the mansion, and 3 weapons.

the question is.. how many kinds of guesses can i make from choosing one character, one mansion room, and one weapon?

what kind of formula do i use for this, nPr or nCr? or can i use the fundamental counting principle?

any help?

2. What you want is $\displaystyle n_1!*n_2!*n_3!$, where $\displaystyle n_1$ is the Character, $\displaystyle n_2$ is the room in the mansion, and $\displaystyle n_3$ is the weapon.

For example: Lets say Professor Plum we know is the murderer. He could have done it in any of the 7 rooms correct? So there are (at the moment) 7 possible scenarios for Plum to have committed the murder. But we also know he can use 3 weapons in each of the 7 scenarios: so thats three weapons in Room 1, three weapons in Room 2, three weapons in Room 3, etc. to three weapons in room 7 - for a total of 21 possible scenarios that Professor Plum (Character), could have gone somewhere (Room) and used a weapon (Weapon) to murder someone. But Plum isn't the only character. We have 5.

Thus you would use the basic methods of counting to find out the total possible scenarios that can play out here.

As an aside, remember what permutations and combinations are: Permutations are just an arrangement of $\displaystyle n$ objects, and combinations are the same arragement with uniqueness factored in.

3. Originally Posted by ANDS!
What you want is $\displaystyle n_1!*n_2!*n_3!$, where $\displaystyle n_1$ is the Character, $\displaystyle n_2$ is the room in the mansion, and $\displaystyle n_3$ is the weapon.

For example: Lets say Professor Plum we know is the murderer. He could have done it in any of the 7 rooms correct? So there are (at the moment) 7 possible scenarios for Plum to have committed the murder. But we also know he can use 3 weapons in each of the 7 scenarios: so thats three weapons in Room 1, three weapons in Room 2, three weapons in Room 3, etc. to three weapons in room 7 - for a total of 21 possible scenarios that Professor Plum (Character), could have gone somewhere (Room) and used a weapon (Weapon) to murder someone. But Plum isn't the only character. We have 5.

Thus you would use the basic methods of counting to find out the total possible scenarios that can play out here.

As an aside, remember what permutations and combinations are: Permutations are just an arrangement of $\displaystyle n$ objects, and combinations are the same arragement with uniqueness factored in.
Ohhh.. so to work it out, it would be 5! x 7! x 3! ??
thanks

4. Originally Posted by oryxncrake
there is a game that involves a murderer, a mansion, and the weapon used to commit the murder. there are 5 characters in all, 7 rooms in the mansion, and 3 weapons.
the question is.. how many kinds of guesses can i make from choosing one character, one mansion room, and one weapon?

what kind of formula do i use for this, nPr or nCr? or can i use the fundamental counting principle?
use the fundamental counting principle: $\displaystyle (5)(7)(3)$.
The question is about content not order.
Therefore no factorials are necessary.