3. Let k, m, n ∈ N such that k < n. Use a shortest-paths argument to find a formula for the number of increasing functions from f : [m] → [n] such that f(x) < x + k for all x. Your answer should have the form (a choose b) - (c choose d)

2.

according to newton’s cannonball model, what happens if the cannonball's velocity is enough to overpower the pull of earth’s gravitystep-by-step explanation:

whatvare the choices

step-by-step explanation:

let's call the solutions x₁ = (5 - 2√7)/3 and x₂ = (5 + 2√7)/3

then the equation is (x-x₁)(x-x₂) = 0

if you fill that in you get a lot of numbers which you can recognize as ax²+bx+c=0.

step-by-step explanation:

you can see that a=1 (there is just an x²)

b = (-5 -2√7 + 2√7 - 5)/3 = -10/3

c = -2√7)/3 ) * (-5+2√7)/3) = -1/3

so your answer is 3x^2 – 10x – 1 = 0