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Daily Magic Spells Nov 1st

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Daily Magic Spells Nov 1st
by Yehui Feng - Wednesday, November 1, 2023, 10:37 AM

Rob is attending a party and was asked to bring 10bottles of soda. There are 4 different flavors available at the store, including grape soda which is Rob's favorite. If Rob wants to make sure to bring at least 3 bottles of grape soda, in how many different ways can he choose which sodas to bring to the party?

For this question, I don't get the solution. First, assume 3 soda is already in grape flavour, then only 7 soda need to be put into 4 different flavours. This means we only need to insert 3 bars. It didn't mention that there is at least one for each flavour, so it has 7+1 places to insert bars. So it should be 8 choose 3 right? But the solution says 10, so choose 3. 

Even if it is like at least one for each flavour, it would be even smaller, 6 choose 3.

So why is it 10 choose 3 ????

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Re: Daily Magic Spells Nov 1st
by John Lensmire - Wednesday, November 1, 2023, 11:48 AM

Good question Yehui!

The answer of "10 choose 3" is correct. Be a little careful with how you insert the 3 bars after we set aside the 3 bottles of grape soda. Note that since it is NOT required to have at least one of the other soda types, it is possible for two of the bars to be placed in the SAME space, so there are actually more than "8 choose 3" outcomes. there.

Think about the 7 (remaining) bottles as 7 stars and we have 4-1 bars. So how many ways can you arrange the 7 star symbols and 3 bar symbols all together?

Note: For a quick refresher on Stars and Bars you can also check out one of our old "Thursday Tidbit" YouTube video explaining Stars and Bars by clicking here.

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Re: Daily Magic Spells Nov 1st
by Yehui Feng - Wednesday, November 1, 2023, 1:16 PM

Oh, I get it now, I am thinking about another theorem of stars and bars, a bit mixed up with them... Thank you!