IhaveOBJD

Since this is an AoPS question, I'll give you a start-up but not the whole solution.

Here, we would use permutations. However, what will be different from the regular question, is one person can hold two positions.

We have a prime factorization of $2^4 \times 3^3 \times 5^2 \times 7^2$

If we add one to each exponent, then add up all the exponents.

$5 + 4 + 3 + 3 = 15$

Therefore there are 15 natural factors.

Hope that helps!!!

🙃🙃👌

4/3 = 1 1/3

The Question says 140% more. That means that we would do 100% + 140%  which is 240%.

What you did is of. Which means we would multiply directly.

Hope that helps!!!!

🙃🙃👌

Total Muffins ÷ 6 = Muffin's per tray

New Amount of Muffins per tray is: Muffin's per tray + 5 which is 20.

20-5 = 15 muffin's in the original tray

15 x 6 = 90

Therefore, she had 90 muffin's in her original batch

Idk rlly lol,

I just extended the line from  A to F and realizes it is a bit similar...

To solve this, we can make a few equations:

For the first triangle on the left

4y -18 + y + 14 + 6x - 11 = 180

5y + 6x  = 180 + 18 + 11 - 14

5y + 6x = 195

For the second triangle on the right

6x - 11 + x + 5  + 81 =  180

7x = 180 - 81 - 5 + 11

7x = 105

x = 15

5y + 90 = 105

5y = 15

y = 3

Therefore y is 3 and x is 15.

Hope that helps!!!!

🙃🙃👌

It may be 4, I am not 100% sure.

Let me know whether that's correct or not.

Cheating in AoPS??

I won't solve it, but I'll give you a helpful start.

Clearly, if we add up everything, then we would overcount, which we do not want. We want to count things once and only once. (Sound Familiar)

Now, make an expression where you add up everyone. The subtract that by two times the people who did all languages. (I technically gave you the answer.) 

Hope that helps!!!

🙃🙃👌