Pascal's triangle and inheritance laws I'm trying to get a grasp on genetic inheritance laws.  Assuming that chromosomes remain intact, would the 23rd row of Pascal's triangle accurately show the distribution of inheritance from a set of grandparents?  The first number would be the occurrences of the 23 chromosomes split 0-23,then 1-22, 2-21 ...22-1, 23-0.
Other familial relationships (grandchild, aunt/uncle, niece/nephew, half sibling) have the same 25% as the average transmission, but the pathways are a little more complicated.  Would the distribution still be the same?
(After reading a comment) To clarify the biological assumptions:  There are no double relationships involved.  While I have 23 pairs of chromosomes, I know that 23 came from each parent.  Each of the 23 chromosomes is distinct, therefore I must look at permutations.  From my father, who got 23 from each parent, I'll say 0=grandpa and 1=grandma.  Position 23 (xy pair) must be 0 for paternal and 1 for maternal sides.  So, I should actually be looking at row 22 instead of row 23 in a Pascal Triangle for the paternal grandparents.  On the maternal side, the x chromosome could have come from either parent, so row 23 should the correct one. 
After doing all of the calculations using n!/x!(n-x)! (using n=23) it struck me that the math is much easier to visualize using Pascal's Triangle or a Galton Box.  When looking at DNA test results I want to be able to explain why we do not always inherit exactly 25% from each grandparent.  In reality, this average is not going to happen, ever, barring an anomaly such as extreme genetic recombination in a biased direction that would result in the equivalent of 11.5 chromosomes from each grandparent.
It's been several decades since I was in college, and I was hoping to confirm that my reasoning is correct before I copy a 20 row Pascal Triangle (the largest I could find) and add rows.  And recalculate everything using n=22.
 A: I would say that under normal circumstances, the mother contributes one chromosome in each of the pairs $1$ to $23$ and the father contributes the other.
Among the $23$ chromosomes contributed by the mother, each one has a 
$\frac12$ probability to come from the maternal grandmother and 
$\frac12$ probability to come from the maternal grandfather.
I assume each of those probabilities is independent,
so the number of chromosomes from the maternal grandmother is a binomial random variable with parameters $n = 23$ trials and probability $p = \frac12$ of success on each trial.
While it is true that one of the pairs of chromosomes gets extra attention because it (usually) determines biological sex, the father has an $X$ and $Y$ chromosome in that pair and a priori (before knowing anything else about a person) we should say that either one is equally likely to be donated (at least to a first-order approximation).
Meanwhile the mother has two $X$ chromosomes, one from the maternal grandmother and one from the maternal grandfather, either of which is equally likely to be donated.
I don't think there's any reason to treat the sex chromosomes differently than the others when computing the probability distribution of the number of chromosomes contributed by each grandparent. Yes, it's relatively easy to tell when a particular individual did not get that chromosome from their paternal grandmother, but if we start worrying about that then we also ought to worry about eye color, blood type, and other traits that can easily be used to eliminate possible inheritance of chromosomes.
By the way, you don't need to complete a Pascal's triangle to compute the coefficients of a binomial distribution.
You can use the fact that
$$ \binom nk = \frac{n - k + 1}{k} \binom n{k-1}. $$
Start with $\binom{23}{0} = 1$; multiply by $23$ to get $\binom{23}{1}$;
multiply by $22$ and divide by $2$ to get $\binom{23}{2}$; and so forth.
