Lesson 1

If you look closely at Pascal's triangle, it is visible that approaching the center of the row (whether from the left or the right: the triangle is symmetric), the values increase. Accordingly, the largest coefficient in each row is located in the center. It turns out this is true in general (and not just for \(0 \le n \le 5\), which we just examined). Before proving this, let's see how the coefficients behave for \(n=10, 15, 20\). To do this, we consider all binary strings of length \(n\) and for each \(0 \le k \le n\) we draw a bar, the height of which equals the number of binary strings that contain exactly \(k\) ones (which is exactly \(\binom{n}{k}\)).

import matplotlib.pyplot as plt
from itertools import product

for n in (10, 15, 20):
    plt.clf()
    plt.hist([sum(s) for s in product((0, 1), repeat=n)], n + 1)
    plt.savefig(f'bincoefdistr{n}.png')