Question

you have 4 stones to put each in one of 3 boxes. Each box has a number on the side of it, 0,1 or 2, which will be applied as the value of the stone. how many combinations are possible for a total value of 5?

Answer 1

are the stones distinguishable?

Answer 2

Sorry, its actually a physics question i tried to re word, you have 4 particles that can occupy one of 3 discrete energy states. \[\epsilon_0 = 0, \epsilon_1= 1, \epsilon_2 =2 \] what are the possible combinations to get 5? yes the particles are distinguishable.

Answer 3

you mean for example you could put 2 stones in the box marked 2 and one on the box marked 1 and one is the box marked 0 for a total of 2 times 2 + 1 =5?

Answer 4

yes satellite, that's what i mean

Answer 5

if we ignore the fact that the stones are distinguishable for a second there looks like there are only two possibilities

0,1,2,2
and 1,1,1,2

Answer 6

then to arrange the two sets we can use the multinomial coefficients

\[\frac{4!}{1!1!2!}\] and \[\frac{4!}{3!1!}\]

Answer 7

Thanks zarkon