Statistics - Permutations & Combinations Confirm Solution/Help?
So I did the first 3 questions without any problems, but I'm stuck on 4.
I first drew what the arrangement might look like and got:
T L I H I L I L
I L I L T L H I
T = all together now
H = hot pink
I = IPA
L = lager
At first I thought I would do 2 * 1 * 3! * 3! because there are 2 options of beer and since the hot pink bottle cannot be adjacent to the all together bottle, it forces hot pink into position four, leaving only that 1 option. The 3! and 3! came from the three positions leftover for each beer type. I thought I would do 2 * 1 * 3! * 3! + 2 * 1 * 3! * 3! because there are two different sides I have to account for but I feel like I'm accidentally counting some positions too many times.
I'm not sure if the 2nd digit should be a 1 either. I would really appreciate some help or if someone told me what's wrong with my solution as I'm very confused as to how I would solve 4). I think I would've been fine if the question only asked for a specific side, but with both the left and right side I'm not sure what to do.
- PuzzlingLv 72 months agoFavorite Answer
Let me go through all the parts.
From the 30 IPAs, pick 4 --> C(30,4)
From the 6 Lagers, pick 4 --> C(6,4)
C(30,4) * C(6,4)
We only care about one particular selection of 8 beers.
We have two alternating patterns (ILILILIL or LILILILI).
For the 4 IPAs we have 4! ways to arrange them.
Similarly for the 4 Lagers we have 4! ways to arrange them.
2 * 4! * 4!
Once we place a beer, that will determine the alternating pattern. For example, if we put an IPA in position 3, then the other IPAs must go in the remaining odd positions. And if the first IPA goes in an even position, the other IPAs must go in an even position.
Let's place the two pink beers first.
If All Together Now goes on an end (2 choices), then there are 3 choices for where Hot Pink goes --> 2 * 3 = 6
If All Together Now goes in a middle spot (6 choices), then there are 2 choices for where Hot Pink goes --> 6 * 2 = 12
So there are 18 ways to place the special pink beers. Then there are 3! ways to place the other IPAs and 3! ways to place the other Lagers.
18 * 3! * 3!
Again, let's place the special pink beers first.
If All Together Now is placed in positions 1, 2 or 3, then there are 2 places where Hot Pink can go on the right side --> 3 * 2 = 6
If All Together Now is placed in position 4, then there is 1 place it can go on the right side (position 7) 1 * 1 = 1
So there are 7 ways to place the special pink beers where All Together Now is on the left and Hot Pink is on the right. Again we have 3! * 3! ways to place the other beers.
7 * 3! * 3!