Alan

Not just 2's, both of them are also divisible by 3.

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This is simply (n+10)!

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Be great if we could!  This is a still unresolved question that no mathematician in the world has yet been able to answer.  Fame (if not fortune) awaits the person who eventually does so!

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$${\mathtt{0.6}}{\mathtt{\,\times\,}}{\mathtt{4}}{\mathtt{\,\small\textbf+\,}}{\mathtt{0.2}}{\mathtt{\,\times\,}}\left(-{\mathtt{1}}\right){\mathtt{\,\small\textbf+\,}}{\mathtt{0.2}}{\mathtt{\,\times\,}}\left(-{\mathtt{10}}\right) = {\frac{{\mathtt{1}}}{{\mathtt{5}}}} = {\mathtt{0.2}}$$

 Expected win = $0.20 

The commutative property says that  x*y = y*x  (for multiplication) and that x + y = y + x (for addition)

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First divide through by 3 to get

x^2 + 16x + 64 = 0

The left-hand side is simply (x+8)^2 so there is a double root at x = -8

Check by drawing a graph of the original quadratic and seeing where it hits the x-axis.

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A Monte-Carlo simulation suggests the probability is:

(1 + 2/3 + 2/3 + (4/3)*(2/3)²)*(1/3)² = 79/243 ≈ 0.325

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Start by finding two numbers whose product is 6.

These are:

1 and 6

2 and 3

Now see if either pair adds to 5. 

Clearly, 2 and 3 do.  

So x^2 + 5x + 6 can be factored as (x + 2)(x + 3)

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Force = mass*acceleration  (F = m*a)

a = 160/15 = 32/3 m/s²

v = u + a*t where u = initial velocity (0), t = time (0.12s), v = final velocity

v = (32/3)*0.12 = 1.28 m/s

(Since the initial velocity was zero, this is the change in velocity).

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