thelizzybeth

The sum of the numbers is 10x10=100. When we exclude a number x from this set, our new average is \(\frac{100-x}{9}\). We are told this equals 9. So our equation is \(\frac{100-x}{9}=9\).

Solve for x and you have your answer. 

Put all the three boys as one block. In the block, they can be arranged in 3x2x1=6 ways. Then the arrangment for four people (the three girls and one block of boys) is 4x3x2x1=24. 

Multiply 6x24 to get our answer, 144. 

Looking for a pattern, we find that the numbers are decreasing in two ways.

First, starting from the very right (the number 9) to the left.

Second, decreasing from left to right starting from the number 5. 

So we can see through this pattern that the next term is 9876541234568.

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If a quadratic has nonreal roots, then the discriminant \(b^2-4ac<0\).

Plug in the values for b, a, and c into this equation to get \(k^2-4*12*27<0\). 

Rearrange to get \(k^2<4*12*27\). 

Now factorize 4*12*27 and solve to get \(-36 < k < 36\)

There are numbers that are missing.

For 1x4 rectangles, the yellow ones, there are 10 of them (going horizontally)

For 4x1 rectangles, the blue ones, there are also 10 (going vertically).

For the 2x2 squares, the green ones, there are 16. 

Add the cases up to get your answer. 

I found these cases by handcounting every rectangle, but if you would like an algebra solution for problems like this with bigger numbers, I could explain that as well.

Factor the top term and bottom to get:

\(\frac{12u^2(u+4v)}{3(u+4v)}\)

Now we can cancel out the term (u+4v) since the problem tells us that this doesn't equal 0 (division by 0 is undefined)

\(\frac{12u^2}{3}\)

Now simplify to get:

\(4u^2\)

Combine like terms to get: 

\(-7ab^2+9a^3b\)