Subtracting Unwanted Combinations in a Fish Store Sale

Question

So, I'm working out one of my assignments and I'm a little bit stuck on this problem:

A fish store is having a sale on guppies, tiger barbs, neons, swordtails, angelfish, and siamese fighting fish (6 kinds). How many ways are there to choose 24 fish with at least 1 guppy, at least 2 tiger barbs, at least 3 neons, exactly 1 swordtail, at least two angelfish, and no more than 3 siamese fighting fish?

I feel like the way to approach this problem is to use r-combinations with repetition (because the fish store isn't running out of any of the fish) and subtract the combinations that I don't want. Here's what I've got so far:

We need at least 1 guppy, 2 tiger barbs, 3 neons, and 2 angelfish. If we set those aside (all 8 of those), we're down to $24-8$, or 16 remaining fish to choose. Add the 1 swordtail, and we only need to choose $24-9$ or 15 fish out of 5 ($(6-1)$ because there were 6 kinds, but we can only have 1 of the swordtail).

All this accounted for, utilizing r-combination with repetition, $\binom{n+r-1}{r}$: $$\binom{15+5-1}{15}=3,876$$

BUT I can't have more than 3 siamese fighting fish! How do I subtract all the groups with more than 3 siamese fighting fish?

Any help would be much appreciated! Also, if I've made any other errors, please point those out and I'll edit the question appropriately. We covered it in class, but it went so quick I didn't get it down in my notes. Thanks!

Answer 1

It seems like you could just sum up all the cases where you have $0$, $1$, $2$, and $3$ siamese fighting fish separately. If there are no siamese fighting fish, then we can choose $15$ out of $4$ types, if there is exactly one siamese fighting fish, then we can choose $14$ fish out of $4$ types, if there are exactly two, then we can choose $13$ fish out of $4$ types, and if there are exactly three then we can choose $12$ fish out of $4$ types. This would give

${15+4-1 \choose 15}+{14+4-1 \choose 14}+{13+4-1 \choose 13}+{12+4-1 \choose 12}$

Answer 2

Here is an approach using generating functions. The generating function for the guppies is $g+g^2+g^3+\cdots ={g\over 1-g}$. In the same way, the generating functions for the other 5 types of fish are ${t^2\over 1-t}$ (tiger barbs), ${n^3\over 1-n}$ (neons), $sw$ (swordtails), ${a^2\over 1-a}$ (angelfish), and $1+s+s^2+s^3$ (siamese fighting fish). The generating function in six variables is $$\Phi(g,t,n,sw,a,s)={g\, t^2\, n^3\, sw\, a^2\, (1+s+s^2+s^3)\over (1-g)(1-t)(1-n)(1-a)},$$ whose coefficients give the counts for all acceptable purchases of fish.

Since we only care that the total number of fish is 24, and not the particular makeup of types, we may look at a simpler generating function where all six variables are replaced by the variable "$f$": \begin{eqnarray*}G(f)&=&\Phi(f,f,f,f,f,f)\\[5pt] &=&{f^9(1+f+f^2+f^3)\over (1-f)^4}\\[5pt] &=&\sum_{j\geq 0}{j+3\choose 3} (f^{9+j}+f^{10+j}+f^{11+j}+f^{12+j}). \end{eqnarray*}

Extracting the coefficient of $f^{24}$ in $G(f)$ gives ${18\choose 3}+{17\choose 3}+{16\choose 3}+{15\choose 3}=2511.$

This isn't as easy as mathochist's solution, but you can use the same idea for more complicated problems.