Hm, the only one I can comfortably help you with is number 3.

I assume you realize this: 12!/7!x5! = 12x11x10x9x8! / 5x4x3x2 (7! takes out 7! from 12!)

After that, let 5x2 take out 10 and let 4x3 take out 12 and you're left with 11x9=99, which is an integer. Hence, no difference from the other two options, which are obviously integers as well.

Hope this helps.

1. For the first problem, you need to drop perpendiculars from the points (-sqrt3,1) and (s,t) to the x axis. You now have two right triangles. Look at the one to the left of the y axis. The legs are -sqrt3 and 1. Those form a 30:60:90 triangle with the 30 degree angle at the origin. The hypotenuse must be 2.

Now look at the triangle to the right. the angle at the origin must be 60 degrees (180-30-90). Because the points are on a circle, the hypotenuses of both circles are radii and there fore are the same length = 2. You have another 30:60:90 triangle, this time with the 60 degree angle at the origin. That makes the x value 1 and the y value sqrt3.

2. This one asks for an approximation. Try factoring 10^2 out of the denominator and the numerator. You get [(10^6) - 1]/[(10^5) - 10]. The one and ten are insignificant compared to 10^6 and 10^5 so for an approximation we can ignore them. We are left with 10^6/10^5 = 10

4. This one takes careful reading. The tree starts at 4 feet and grows x feet per year. We are told the it is 20% taller after year six than after year four. WE can now set up an equation:

4+6x = 1.2 (4+4x)
4 + 6x = 4.8 + 4.8x
1.2x = .8
x = .8/1.2 = 2/3

Can you list out the answers?

#1
The answer for the first should be sqrt3.
The length of the lines to both points are equal, as well as their angles from the x-axis itself.

#2 you can factor out 10^2 from top, 10^3 from bottom and you can reduce quite a bit. In the end you'll end up with

(10^ 6 - 1) / (10^ 5 - 10)

i thought that approximates with 10. The -1 and -10 would only contribute meaningfully if the numbers were small. for example, (x-1) / (x-10) where x = 11. Then you get 10. But if x is 1,000,000 then you get 999,999/999,990 you get something close to 1.

In this case the numbers are so large that it trivializes the subtracted values

so the answer I think is 10

#3 as explained by the other poster, you can factor the first two and end up with a partial factorial on the numerator. The 3rd one, you can similarly factor out and end up with a 1 in denominator.

#4 The tree grows by a constant amount each years

4 + x + x + x + x + x + x = height after 6 years
4 + x + x + x + x = height after 4 years

height after 4 years and 1/5 more = height after 6 years
(height after 4 years)*6/5 = height after 6 years. 6/5 = 1 1/5 which is basically 120% or 1/5 more
(4+4x)*6/5 = 4+6x
24/5 + 24x/5 = 4+ 6x
4 4/5 + 4 4/5 *x = 4 + 6x
4/5 + 4 4/5*x = 6x
4/5 = 1 1/5 * x
4/5 = 6/5 * x
4 = 6x
4/6 = x
2/3 = x

IN the first problem the answer is 1. The angle at the origin is 60 degrees, not 30 degrees. You have a straight angle along the X axis with 30 to the left, 90 in the middle and 60 to the right. The side opposite the 60 degree angle will be sqrt3 and the side along the x axis will be 1

Thanks Tani, for correcting #1, another thing to remember (30,60,90 triangles)

30:60:90 triangles are crucial as well as 45:45:90 triangles. They show up all over the place and you often can't solve a problem without them. The other big triangle problem time saver is the set of Pythagorean triplets.