There is this random walk type of question:
If five H's and five T's are arranged at random, what is the probability that at some point in the sequence the difference between the number of H's and the number of T's up to that point will be three or more?
So I have no idea why the answer is the way it is. Can anyone elaborate?
The number written in the corner of each square, at coordinate $(x,y)$ along the horizontal and vertical axes, is the number of ways that $x$ coins selected from $5$ Hs and $5$ Ts can be arranged so that as you count from left to right, the magnitude of the difference between the number of Hs and the number of Ts is never $3,$ and after $x$ coins you have $y$ more Hs than Ts.
For example, at $(5,1)$ the number $9$ is written because there are $9$ ways to have $1$ more H than T after $5$ coins without ever having three more of one kind than the other.
The number at $(x,y)$ is the sum of the number at $(x-1,y-1)$ and the number at $(x-1,y+1),$ because we get to $(x,y)$ either by adding one H to one of the arrangements that get to $(x-1,y-1)$ or by adding one T to one of the arrangements that get to $(x-1,y+1).$ Of course the number at $(x,2)$ is the same as at $(x-1,1)$ since nothing that goes to $(x-1,3)$ counts as "never having three more of one kind," and similarly the number at $(x,-2)$ is the same as at $(x-1,-1).$
The numbers at the vertices of squares are analogous to the numbers in Pascal's Triangle, except that in Pascal's Triangle there is no rule that says you cannot be more than two steps away from the center line at any time. Conversely, the numbers in the diagram are computed like the numbers of Pascal's Triangle, except that we are not allowed to put a number more than two steps away from the center line.
The number $\binom{10}{5} = 252$ is the number that would have occurred at coordinates $(10,0)$ if the diagram had been constructed like Pascal's Triangle, that is, it is the number of ways to arrange the $10$ coins without the restriction that you can never have three more of one kind than the other.