[Combinatrics] Anagram method vs Formula?

[SK191]

Question: Let us say there are five flavors of icecream: banana, chocolate, lemon, strawberry and vanilla.
We can have three scoops. How many variations will there be?

Formulaic solution:

(5+3−1)!/3!(5−1)! = 35

I understand this solution.

However, when i attempt to solve this using the anagram method, i only get 25 rather than 35. My (incorrect) solution:

Attempted anagram method solution:

YYYNN (3 diff scoops)
YYNNN (2 same, 1 diff)
YNNNN (3 same)

Add up all three: 5!/(3!x2!) + 5!/(2!x3!) + 5!/(1!x4!) = 25

Interestingly, when i change the question to selecting only 2 scoops rather than 3. Both the solutions match. Is this the anagram method's limitation or am i doing something incorrect.


EDIT: After further investigation, i managed to get the correct answer by considering the triple scoop and double scoop as additional two flavours.

Then, the anagram becomes:

YYYNNNN [last two No's for Triple scoop (all same scoops) and Double scoop (2 same + 1 Diff)]

But i don't know why this works.

[tim]

Your anagram method undercounts the combinations where there are two scoops. Remember, Chocolate Chocolate Banana is different from Chocolate Banana Banana, but you've only accounted for it once. Double that second term and you will get 35. This undercounting won't show up in your two-scoop example because it precludes the possibility of having two or more different numbers of two or more different scoops.

Incidentally, your YYYNNNN reformulation matches with the original formula you used, so you've effectively turned the anagram method back into the formulaic approach. I'd say *that's* why it works, if I knew where your original formula came from. Where did you get that formula, and how is it used? It *looks* like a classic stars and bars argument, but that is a super complicated approach that will never be required for a GMAT problem (which is why it doesn't show up in our strategy guide). Stars and bars is very powerful and *can* be used sometimes on the GMAT (just like I could overkill a GMAT geometry problem with calculus), but you have to be *really* careful that you know exactly what you're doing, because the risk of misusing that formula outweighs its potential benefit unless you are 100% sure you are applying it correctly.