1. ## Trivial Preoccupations.

I'm throwing a big BBQ in my backyard, and I need some help gathering foodstuffs. By my calculations, I will need 8 tubs of ice cream to feed everyone for dessert, and I send you to the supermarket to purchase some. Coincidentally, the supermarket sells exactly 8 different flavors and has an abundance of each. If my only instructions are for you to return with exactly 8 tubs i.e. I don't care if you repeat flavors at the sacrificing of others, with how many different groupings can you possibly return?

2. Hello, Bilbo Baggins!

I'm throwing a big BBQ in my backyard, and I need some help gathering foodstuffs.
By my calculations, I will need 8 tubs of ice cream to feed everyone for dessert,
and I send you to the supermarket to purchase some.
Coincidentally, the supermarket sells exactly 8 different flavors and has an abundance of each.
If my only instructions are for you to return with exactly 8 tubs,
i.e. I don't care if you repeat flavors at the sacrificing of others,
with how many different groupings can you possibly return?

For each purchase, you have 8 choices of flavors.

Since you make 8 such decisions, there are: . $8^8 \,=\,16,\!777,\!216$ possible outcomes.

3. Originally Posted by Soroban
Hello, Bilbo Baggins!

For each purchase, you have 8 choices of flavors.

Since you make 8 such decisions, there are: . $8^8 \,=\,16,\!777,\!216$ possible outcomes.

Thanks for your reply. However, you have misinterpreted my question. You are using an approach that I call "product of options" for sequence-specific, repetitive grouping. And you are correct. Had I stated that the sequence of flavors is of consequence, your approach would yield the correct answer. However, sequence has no relevance here. I don't care what order you put them in your bag at the grocery store or what order you put them down on the kitchen table when you return. I merely care about their grouping.

For example, by your approach, you're contending that ABC and CBA are two distinct outcomes. For this problem, I'm only interested in A, B, and C being together in one group and would regard these two groups as being the same.

(I actually know the answer to this; I'm just curious to see if someone can answer it.)