@readingallday Absolute value is a number's distance from 0. This is why positive numbers stay positive, and negative numbers become positive in these problems, as we are just counting how many units a number is from zero. Ex: (-7 is 7 units away from 0 & 7 is also just 7 units away from 0 but in opposite directions).

@JoyfulSapling That's actually really cool

and that's why you can't divide by zero 🙂 if you divide both sides by (5-5) its dividing by zero, which breaks math

@punctualmoose PAB is similar to PCD because

They both share angle P. They share two sides, therefore angle PAB = angle PCD and angle PBA = angle PDC.

Explanation:
Of the 6 cookies Pat chooses, there are three types: chocolate chip, oatmeal, and peanut butter cookies. We also note that there can be 0 of a type. We can use the stars and bars formula to divide 6 objects into three categories (n is the number of objects, which is cookies in this case, and k is the number of categories):

n+k-1 choose k-1
= 6+3-1 choose 3-1
= 8 choose 2
= 28

That is why the answer is 28.

Please find below the proof for the stars and bars formula.
Let's use the cookies problem from above to help us with the proof. Right now, n=6 (6 cookies) and k=3 (3 types of cookies).
Putting 6 cookies into 3 categories looks like this:
costume1.png
There are 8 objects total in that picture (6 cookies + 2 bars).
Let's say we remove all objects and hold them. There are 8 spots to put our 8 objects. We can choose 2 spots of these 8 spots to put our bars, and the cookies go into the remaining 6 spots. Therefore, there are 8 choose 2 = 28 different combinations of 6 cookies.

So, in the formula n+k-1 choose k-1, n+k-1 represents the total number of spots to put the objects and the bars, and k-1 represents the number of bars needed to split the objects into k categories (for example, in our problem, 2 bars were needed to split the 6 cookies into 3 categories).

it's fixed now, nvm

111,111/11 = 10101
111,111/13 = 8547

@ingeniousnewt 12 squared is 144, and 18 squared is 324

@mathnerd_101
Yes you did thank you.

@sophisticatedwolf Typically, when posed with these problems, you will consider the positive number only, unless otherwise specified. The reason is because who actually cares about negative integers

@MS8609 Great, I'll try that out, thank you for your suggestion!