ExpireNight

The set S contains 72 numbers. Since the restriction is that the first term must be a prime, only 2, 3, and 5 work.  3 * 4 * 3 * 2 * 1 = 72

Now, if we need the third term to equal 2, we can get 2 * 3 * 1 * 2 * 1, which equals 12. 

12/72 = 1/6

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Hi!

So we need to make 3 letter sequences. Looking at your question, there doesn't seem to be any restrictions on what the first letter can be, so we have 26 choices for the first letter. The second letter must be a vowel, so we have 5 choices. Finally, the third letter has to just be different from the first letter. We have 25 choices because it just needs to differ from the first letter. 

26 * 5 * 25

3250 sequences can be formed!

Hi!

First, we say that the number of students in class c = c, class b = b, and class a = a. 

We can see that Class B has 50% more students than Class C. The ratio between them is 1.5:1, or 3:2(Class B: Class C). 

Then we can look for the ratio between Class B and Class A. Class A also has 50% more students, making the ratio between them also 1.5:1, or 3:2(Class A: Class B).

Now we can look for the ratio between all three of them. Class B to Class C is 1.5:1. We can then multiply 1.5 by 1.5 again to get 2.25 as Class A.

2.25:1.5:1

2.25:1

9:4 is the ratio.

To check, we can pretend Class C has 8 students. Class B then has 12 students, and Class A has 18. 

18:8

9:4

Hi! 

If we say that x is the number of plain cookies, then 6x/7 is the number of raisin cookies. If we add x and 6x/7, we get 13x/7.

13x/7 = 104 

13x = 728

x = 56

We have 56 plain cookies and 48 raisin cookies. Now we need to take a certain number away from both of these numbers to get a ratio of 3:5. 

(48-  x)/(56 - x) = 3 / 5

(48-  x)5 = (56 - x)3

240 - 5x = 168 - 3x

72 = 2x

36 = x

Now that we have this number, we can find out that there are 48 - 36 = 12 raisin cookies left.

Sorry, the question is:

How many seating arrangements are possible?