Deduce equation for a particle moving a particular distance

Question

Solve the following equation:

Suppose that a particle, starting at the origin, has an equal chance of moving to the left or right by a distance ?x in a time interval of ?t.

(a) Let n>0 be an integer, and let m be an integer, such that -n≤m≤n and n-m is even. By computing the number of ways that the particle can move a net distance of m?x in n time intervals ?t, show that the probability that it is at x = m?x, after a time t=n?t is

     n!(1/2)ⁿ)/(1/2(n+m))!(1/2(n-m))!                                 (1.1)

(b) Use Stirling's formula ≈ , for large n, to deuce that (1.1) is approximately

n!≈ √2πe^-n n ^n+(1/2)

2/√2πt(1-(x/t)²(Δt/Δx)²)^{(-t/2Δt)-1/2 (1+(x/t) (Δt/Δx))(x/Δx)-1/2   (1-(x/t)} (Δt/Δx))^(x/Δx)-1/2 (Δt)^1/2

(c) Note that we get a well-defined density of order Δx , if Δt is proportional to Δx^₂ , say = (Δx)²=2kΔt. Then dividing (1.2) by 2Δx and letting  Δt→0, show that we obtain the fundamental source solution

(e-^x2/4kt)/√4πkt