First, about free combination, why should we multiply the same gene coefficient? This is a probability problem in high school mathematics. The probability of two independent events happening at the same time increases exponentially. That is, P(AB)=P(A)P(B)

Second, there is no need to self-cross, because homozygotes self-cross, and the offspring genes are not separated. For example, the offspring of AA are all AA, and the probability of its occurrence is 100%, that is, the coefficient before selfing is 1. Probability theory +0× 1 = 1

Third, everything has a certain probability. For example, the probability of a coin getting the word side at a time is half. Similarly, in free mating, the probability of Aa passing through gametes A and A is half.