Permutations & Combinations order and adjacency related questions

[Eagerissac]

Struggling with c and d and was hoping someone could help. I wrote my attempts for a and b as well and would really appreciate some confirmation

Scott’s Dairy Parlor sells 40 types of ice cream and 8 types of frozen yogurt.

a.The owner of Scott’s Dairy ParIor is required to choose 3 types of ice cream and 3 types of frozen yogurt. CaIculate the totaI number of options that the owner has.

I did 40C3 * 8C3 = 553280

b. The 6 frozen dessert items (3 ice cream and 3 frozen yogurt) chosen in a) are required to be positioned in a Iine on a display rack so that no two ice creams are adjacent and no two frozen yogurts are adjacent. CaIculate the totaI number of ways of doing this.

2 x 3! x 3! = 72

c. Continuing from b), suppose that two of the desserts selected were Cotton Candy (an ice cream) and Blueberry (a frozen yogurt). Due to the reason that both of these desserts are the colour blue, the owner does not want to place them adjacent to each other. CaIculate the total number of ways there are to do this (whiIe stiII aIternating between ice cream and frozen yogurt).

This one I feel like I'm wrong but I basically took 72 from b) and subtracted 2 x 3 x 3 because there are 2 ways of doing it, and 3 different spots to place the blue ice cream as well as the blue frozen yogurt. 72 - 12 = 60.

Someone told me I'm supposed to multiply '5 x 2 x 3 x 3' and then subtract but I have absolutely no idea where the 5 came from. If they're right, could someone try to illustrate maybe? I thought it'd be maybe 6 but 5? I can't see where that came from.

Or if there is another way of soIving this, pIease show me l'd reaIIy appreciate it!


d. How many of the arrangements from the previous question have the Cotton Candy fIavor among from the 3 Ieftmost containers and the container of the BIueberry flavor among the 3 rightmost containers?

This one I'm a little at a lost for what to do. My attempt was following the above solutions and doing 2 x 3! x 3! and then also multiply them by 2 cause there are at most 2 spots on each half of the shelf that the blue frozen yogurt and ice cream can be place on?

2 x 3! x 3! x 2 x 2 = 288.

 

[Romsek]

a) is correct
b) is correct

for (c) Let's place the other flavors first. There are 8 ways to arrange the non-blue flavors so they alternate.
Now we can insert the 2 blues adjacent to one another in 5 slots, and they can be ordered only way within this slot depending on the adjacent types.
That gives us 40 ways, 8x5 that the two can be arranged together, and thus 32 ways in which they won't be.

for (d) the trick is that if CC is in the first 3, then B cannot be in the first three. Similar for B being in the last 3.
So it's really just a question of how many ways can CC be in the first 3. Well it's just as likely to appear in the first 3 as the last 3 so it will be half of the arrangements, i.e. 16.