E.g., binomial expansions: (p + q) k
p = probability of event happening
q = 1 - p (probability of not happening)
k = sample size (number of trials)

the coefficient indicates the number of ways an outcome may occur the number of each term is the probability of that outcome, e.g.,

for k=3 ----> p3 + 3p2q + 3pq2 + q3 = 1
three coins TTT 3HTT 3HHT HHH
five coins:
TTTTT 5HTTTT 10HHTTT 10HHHTT 5HHHHT HHHHH

More possible outcomes but lower probability of each.