BuilderBoi

\(-{1 {2 \over 3}} = -{5 \over 3}\)

\((-{5 \over 3})^2 = -{5 \over 3} \times -{5 \over 3} = \color{brown}\boxed{25 \over 9}\)

\(99 \times 99 = (100 -1)(100-1) = 100^2 - 2 \times 100 + 1 = \color{brown}\boxed{9801}\)

9 mod 6 is 3, not 6.

\(4^{10} = 2^{20} \)

\(8^6 = 2^{18}\)

\(2^{20} \div 2^{18} = 2^2 = \color{brown}\boxed{4}\)

After we put 1 of each action figure in each box, there are 3 Scuba Steves and 2 Diving Dans left.

Using stars and bars for the number of Scuba Steves gives us 3 stars and 3 bars, for \({6 \choose 3} = 20\) combinations

Doing the same thing for the Diving Dans gives us 2 stars and 3 bars for \({5 \choose 3} = 10\) combinations.

So, there are \(20 \times 10 = \color{brown}\boxed{200}\) ways to do this.  

That's actually a really cool formula!

I got a different answer, Melody.

If a number is both a square and a 5th power, it must also be a perfect 10th power.

Note that \(1024^2 = 4^{10}\), so there are 4 numbers that work: \(1^{10}\), \(2^{10}\), \(3^{10}\), and \(4^{10}\)

There are \({4 \choose 2} = 6\) successes and \({1024 \choose 2} = 523776\) total outcomes.

So, the probability is \({6 \over 523776} = \color{brown}\boxed{1 \over 87296}\)

Use stars and bars like proyaop said: 

There are 4 stars (balls) and 3 - 1 = 2 bars.

So, there are \({4 + 2 \choose 2} = {6 \choose 2} = \color{brown}\boxed{15}\) ways.

The inradius of the triangle is \(\text{Area} \over \text{semiperimeter}\)

The circumradius of the triangle is \(\text{product of 3 sides} \over 4\times\text{Area}\)