Math Is Fun Forum

Hi guys,
I hope everyone's been well.

Could someone explain how they figure out these 4 problems? I would truly appreciate it.

1. How many ways are there to choose one-half dozen donuts from 9 varieties so that there are exactly 4 glazed?

2. How many distinct ways are there for arranging the letters in “arranging” so that the vowels are together in 
any order?

3. A box of 10 RAM chips contains three defective and seven good chips. Three chips are chosen at random without replacement from this box and tested. What is the probability that exactly one of the chips is defective?

4. On a given day, there is a 0.7 probability that it is sunny. Jane walks past an ice cream stand every day. When it is sunny, the probability that she buys an ice cream is 0.8. When it isn't sunny, the probability that she buys an ice cream is 0.1.
What is the probability that Jane buys an ice cream?

If someone could, I just don't understand how you set up these double integrals. I know it involves using r and such, but I keep messing up the final computation. If someone could just go steo-by-step through each problem and explain how you set them up, I think I can compute the actual double integral. I just need to understand how to start off the equations. I had to split up the pages by screenshots, so I apologize if one or two of them seem cut off. All pages should continue to the next page. I just really, really could use some help in understanding these problems. Thank you so much!

https://screencloud.net/v/6oMd
https://screencloud.net/v/Fp3y
https://screencloud.net/v/iaT3

Hi guys,
I'm studying for this upcoming test. In one of the examples when they find f(x,y) for the function

They differentiate with respect to x first and get

then they simplify it to

Can someone please explain how they reached this step? I'm utterly confused over how to simplify it to that form. I know the y in the numerator comes from the y in the original function, so I guess my question is how they simplified the denominator so elegantly. Thank you very much!

Hi guys I'm trying to understand these multiple differential equations and their solutions for an exam. If it's okay, I posted the solution to the example and I have a question: "In the example it says "Thus, 

  ,

, and so

and
How did he use the two equations to reach the conclusion that

  ? I feel like I'm missing something obvious, and I feel so dumb. Anyways, thanks guys so much for your help, any input would be appreciated.

Here's the problem (sorry it cut off text in one of the pictures, luckily I'm only concerned with the solution)

Hi guys, so I'm working on this problem, and I think I'm (hopefully) close to a solution and I'm just messing up when it comes to simplifying everything. Of course, since this is my own homework, I'm not asking for an answer, just simply a hint or telling me what I'm doing wrong . The problem is this:

   

(I don't know if it will show or not but the problem is the integral from 1 to 2 of 5ln(4x)dx)
So I used

then I said

And when I integrated I got the answer

I factored out a 5x for to get

Then I plugged in the bounds and got

The ones cancelled out and then I factored out a 5 to get

I forget how but I thought I could reduce it to

but it was wrong 

Could someone please guide me to the correct answer? Am I integrating wrong or condensing the logarithms incorrectly or something else?