Hey guys... :
1) thomas has 2 Cd's and two tapes. in how many arrangements can he play the CD's and tapes one after the ther such that each is played exactly once and one CD is not played immediately before or after another CD?
I think the answer should be 8, but it is 12...
2) if one pint of milk with 2% fat is added to half a pint of cream with 50% fat, what is the percentage of fat in a mixture?
the answer is 18, but I think it should be 26...
Okay there are 4! = 24 ways to choose the order you play the 4 CDs (4 for the first, 3 for the second, 2 for the third, 1 for the fourth). You need to know how many of those put the 2 CDs next to each other and then subtract that from 24 to find the number of ways to NOT put 2 CDs next to each other.
There's a few ways to do this but I would consider these three...
1. You have 4 spots and you want to choose the 2 spots to play the CDs. So (4 choose 2) = 6 which means there are 6 ways to choose the 2 spots out of 4 that you'll put the CDs. Of those 6 ways, 3 of them stick the CDs next to each other (CCTT, TCCT, TTCC). Since 1/2 of the ways put the CDs next to each other, 1/2 of the 24 ways put the CDs next to each other. That means the other 1/2 do NOT, which is 12 (by the way this is the answer, not 8).
2. Think of the CDs as one unit. So now you have 2 separate tapes and 1 double CD. That's 3 objects. How many ways can you order them? You do (3 perm 1) = 6 (T1 T2 CD) (T2 T1 CD) (T1 CD T2) etc. But remember that there are really 2 CDs and that you can put either one of them first. So there are actually 6*2 = 12 ways to stick them together. That means there are 24 - 12 = 12 ways to NOT stick them together.
3. If your brain just fails and you have some time, list them out. It shouldn't take long for a problem like this.
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"2) if one pint of milk with 2% fat is added to half a pint of cream with 50% fat, what is the percentage of fat in a mixture?
the answer is 18, but I think it should be 26..."
I think you missed the fact that there is a half a pint of cream and a full pint of milk. With 1 full pint of milk at 2%, that's 0.02 pints of fat. With half a pint of cream at 50%, that's 0.25 pints of fat. So you have 0.27 pints of fat and 1.5 pints total. That makes 0.18 = 18% fat.
