*permutation /combination help

[jw_gino]

Hello,

I can do this question by writing out the combinations but am not too sure how to do this using the combo formula..

In a family of 3 kids, parents agree to bring the kids to the pet store and allow ea. kid to choose a pet. this store sells only cats, dogs. and monkeys. if ea. child chooses exactly 1 animal and if more than one child can choose the same kind of animal, how many different arrangements of animals could the family leave with?

the answer is 10

[RonPurewal]

hi,
per the forum rules (please read them if you haven't yet!), please do the following:
* write out the ENTIRE PROBLEM, WITH ANSWER CHOICES
* cite the ORIGINAL SOURCE OF THE PROBLEM
if you don't do these things, we'll have to delete the problem.

in any case, this is not the best-written problem in the world. in particular, there is no attempt whatsoever to define the word "arrangement", an issue that makes the problem unacceptably ambiguous. (if you have the same animals but they are chosen by different kids, is that the same "arrangement"? we don't know.)

please post the original source within the next 7 days, or else we'll have to kill the problem. thanks.

[jw_gino]

Original source for the question: mcgraw-hill's gmat (2011 edition)

(how would u approach this question below without simply listing the combos?)

question: in a family with 3 children, the parents have agreed to bring the children to the pet store and allow ea. child to choose a pet. this pet store sells only dogs, cats and monkeys. if ea. child chooses exactly one animal and if more than one child can choose the same kind of animal, in how many different arrangements of animals could the family leave with:

a.6
b.8
c.9
d.10
e. 12

[tim]

i agree with Ron that this question is poorly worded and not like anything you'll see on the GMAT. please understand that using practice questions from dubious sources can actually hurt your performance, and be careful which problem sources you use..

that said, this is a typical "stars and bars" problem. say you have three "stars" representing three pets coming off, say, a conveyor belt:

* * *

the conveyor is designed to churn out first cats, then dogs, then monkeys, and to change the type of pet it churns out you must throw a switch, represented by a "bar". so let's say i want one cat and two dogs. well, i have to throw a switch after the first pet (a cat), and not throw the switch again until after the two dogs come off the line:

* | * * |

what if i want nothing but monkeys? i have to throw the switch twice before any pets come off the line:

| | * * *

one of each?

* | * | *

as you can see, the placement of the bars (throwing the switch) in relation to the stars (pets that have come off the line so far) will determine what "arrangement" of pets we get, and vice versa. so the question of how many arrangements there can be just boils down to a combinatorics problem of fitting 3 stars and 2 bars into a 5-character string: 5!/(3!2!) = 10