thelizzybeth

Thank you!

We first put one ball in each box. Now we have 2 balls to distribute in 4 boxes. Since they are both identical, the cases are

3 1 1 1

2 2 1 1

So there are 2 possible distributions.

We have two options to consider: If the 0 that is removed is the middle digit or if 0 is the last digit.

Case 1: N=a0b

10a+b=(100a+b)/9

90a+9b=100a+b

8b=10a

4b=5a

b=5, a=4

Case 2: N=ab0

10a+b=(100a+10b)/9

90a+9b=100a+10b

10a=-b

The only answer for this is when a and b equal 0. But this number isn't a valid three digit number. 

So our answer is 405

This equals \(x=\sqrt{3+x}\)

Square both sides \(x^2=3+x\).

Then factorize \(x^2-x-3=0\).

Since this expression is clearly positive, x=\(\frac{1+\sqrt{13}}{2}\)

This graph is a circle with center (3,5) and radius 4.

The radius of the circle is still going to be the same after reflection, but the center changes to (3,-1).

So our equation is \((x-3)^2+(y+1)^2=16\).

Expanding this, we get \(x^2-6x+y^2+2y-6=0\).

\(B=-6, D=2, F=-6\)

Add these up to get your answer.

Sorry, that's incorrect. Can you expand more behind your reasoning so I can attempt your method myself? 

First off, just by glancing at the shapes, we can tell that C is different.

A, B, and D's lines are all straight, while C has four curved lines.

So we can reasonably guess that C is the correct choice.

However, you can also test it out. It becomes obvious after a few trials that no matter what you do, it's impossible not to retrace the same edge. 

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The measure of angle BAD is 90 degrees. The measure of EAD is 60 degrees.

90-60=30 degrees

I started this problem by noting that there has to be at least a 13.

1*2*3*4*5*6*7*8*9*10*11*12*13

This only has 2 5's so we go up to 15.

1*2*3*4*5*6*7*8*9*10*11*12*13*14*15

Not enough 2's so we use trial and error to find

1*2*3*4*5*6*7*8*9*10*11*12*13*14*15*16= 16!

Anyone can sit in the first seat so there are 6 choices.

For the second seat there are 3 choices- the opposite gender of the person in the first seat.

For the third seat, there are 2 choices- the other two people of the same gender as the person in the first seat.

For the fourth seat, there are 2 choices- the other two people of the same gender as the person in the second seat.

For the fifth seat, there is 1 choice left- the same gender as the first and third seat.

For the sixth seat, there is 1 choice left- the same gender as the second and fourth seat.

So 6*3*2*2*1*1=72

Leave the 1 out for now. Combine -2/7 and 4/49. This equals -10/49. Combine -4/49 and 16/2401. This equals -40/2401. 

This forms an infinite geometric series with first term -10/49 and common ratio 4/49. (You can go on a few more times to see that it is in fact a geometric series).

The sum is \(\frac{-10/49}{1-\frac{4}{49}}=\frac{-2}{9}\). We add this to our beginning 1 to find our final answer 1-2/9=7/9