Alan

The gamma function is a sort of continuous version of the factorial function.  

When n is an integer then Γ(n) = (n-1)!

But Γ(n) is also defined for non-integer values of n.

.

Π is the Greek upper case letter, pi, and stands for the product (analogous to Σ for the sum).

. 

Possibly as:

$$\prod_{n=2}^K(2n-1)$$

or as:

$$\frac{2^{K+1}\Gamma(K+\frac{1}{2})}{2\sqrt{\pi}}$$

where Γ() is the Gamma function

$$\Gamma (t)=\int_0^{\infty}x^{t-1}e^{-x}dx$$

.

a + 9 = 8[a - (2 - a)]

a + 9 = 8[a - 2 + a]

a + 9 = 8[2a - 2]

a + 9 = 16a - 16

25 = 15a

a = 25/15 → 5/3 → 1.666...

.

The tangent of the angle is given by 14/18, so the angle is tan^-1(14/18).

$$\underset{\,\,\,\,^{\textcolor[rgb]{0.66,0.66,0.66}{360^\circ}}}{{tan}}^{\!\!\mathtt{-1}}{\left({\frac{{\mathtt{14}}}{{\mathtt{18}}}}\right)} = {\mathtt{37.874\: \!983\: \!651\: \!098^{\circ}}}$$

Angle of elevation = 37.9° to the nearest tenth of a degree.

.

To what does the 3/4 refer? (radius? diameter? circumference? other?) Inches? Centimetres? ...?

.

Results are:

(a) 120

(b) 7/15

(c) 1/3

.

Think of the first summation as the position of the first ball.  It can only go into boxes 1 to 7, because the 2nd ball will be in another box to the "right" of it, and the third ball will be to the "right " of that.

It might have been better to write the summations as follows, where each summation refers to each ball:

.

Here's a picture Melody:

The copper colour is the z = 2x² - 3y² surface.  The red blob is the point A = (2, 1, 5).  The blue mesh is the plane tangential to the surface at A (z = 8x - 6y - 5).  

The black line is the unit length vector lying in the tangential plane in the direction of steepest descent.

I took Bertie's solution for the vector in the horizontal plane, found where the end of this would meet the tangential plane, scaled the resulting vector from A to this point to be of unit length and plotted the black line to represent the unit vector in the tangential plane.

The black line goes from (2, 1, 5) to (1.92, 1.06, 4.01)  (approximately).

.