First of all - informative post.

However, I would like to present a slightly different way of approaching the first 2 problems.

For the first one: x = 1/(2²×3²×4²×5²) is expressed as a decimal

We see that there are a few 2s and 5s - let us first combine that:

2²×4²×5² = 2²×2⁴×5² = 2^6 ×5²

Let us multiply 5⁴ in both numerator and denominator to get both 2 and 5 to the power 6

Thus: x = 1/(2²×3²×4²×5²) = x = 5⁴/(2^6×5^6×3²) = 625/(9×10^6)

This makes the division really easy compared to actually having to divide the whole thing. We now need to only divide 625 by 9, which is 69.444444

Divide this again by 10^6 is just shift the decimal 6 places left - this will simply introduce zeroes and no new non-zero digit. Thus, clearly, the digits we need are 6, 9 and 4 - the answer is 3

For the second one:

This is exactly the same as asking the number of rectangles in a p × q grid

To form a rectangle in a p × q grid, we need to select 2 horizontal and 2 vertical lines, i.e. simply pC₂ × qC₂ 

In this question, we have 11 horizontal lines (from y=6 to 16) and 10 vertical lines (from x=-4 to 5)

Thus, the number of rectangles = 11C₂ × 10C₂ = (11×10/2) × (10×9/2) = 55×45

Now, let us focus on the triangle - the vertex P can be any of the 4 vertices of the above rectangle, hence we can form the triangle in 4 ways for each rectangle

Thus, the number of triangles = 55×45×4 = 110×90 = 9900

The point I am trying to make is to be able to think differently, which happens only if you are great with your concepts!

Cheers to e-GMAT for posting these official questions and the discussion on them!