(harder version)

 
It's the end of the school year where we live and time for our annual "school's out" celebration. I asked my nephews and niece what they wanted to do and they all agreed that they wanted me to pick them up on the last day and take them to the nearest fast food restaurant for lunch. So that is exactly what I did. On the way to the restaurant we discussed what we would order for lunch.

"They have a cool new kind of milk shake I've been wanting to try for dessert," explained Danny. "You get to choose any two flavors of ice cream from 7 choices, and they put them both in the glass at the same time, so you get a 'half and half' shake!"
 

"Wow! I've never heard of that, " I said. What are the flavors?"

"Well," said Barney, checking them off on his fingers as he named them, "They have vanilla, chocolate, strawberry, peach, mocha, banana, and chocolate mint."

"I'm having 'straw-bana' for mine," said Gina, laughing at the funny sound of a shake made from strawberry and banana ice creams.

"I'm trying to figure out how many different 2 flavored milkshakes I could create using those 7 flavors," said Barney.
 

"Oh, I already know" squealed Gina. "28!"

"Gina, how do you know that? " I asked.

"I heard it on a commercial the other night so it must be right," said Gina, "But now I am not so sure. Maybe it was 24, or 21. I forget now."

"Hmm. I wonder if there is a way we could prove whether it was 28 or not?" I pondered.
 

"And I wonder how many more ways there would be if we could have both sides of the shake contain the SAME flavor?" wondered Barney. "Uncle Bill would probably want his that way."

"Well, maybe we could figure all this out - but could we wait until AFTER we have our milkshake?" begged Danny. "My brain is on vacation!"
 

How about you? Is your brain on vacation or are you up for the challenge of figuring out how many different combinations of 2 different flavored shakes can be made if there are 7 different flavors? Can you name them all? What if the shakes didn't HAVE to have 2 different flavors? How many ways then? Is there a way to make sure you have them all WITHOUT having to name them all?
 

I can't wait for you to try this summertime math "treat". When you think you have the complete answer, post your answer and solution strategy by clicking on "I want to send in my solution" at the bottom of this challenge. (To avoid the dreaded "How did you get your answer?" response from me, remember to include both your answer AND how you arrived at it.

Then, check back on this solution page in a few days. Just click "I want to send in my solution" again and scroll down until you find your posting. You should find a message from me, Aunty Math, and perhaps from some other people too!

Love,
Aunty Math

P.S. My favorite 'half and half' shake is peachilla. Can you figure out which two flavors I chose? Which two would you choose and what would you call your creation?