Dear All,

I am considering maximal subgroups of odd index in Alternating and Symmetric groups, and this leeds me to some questions on binomial coefficients that I presently do not know and that I need some hints and suggestions :

Assume that $ab=2^t$ with $a>1, b>1,t\geq 3$. Which values of $a$ and $b$ such that $$\frac{(2^t)!}{(a!)^b.b!}$$ is odd and smallest.

Let $n\geq 9$ be an odd integer (not prime) and $p$ the largest prime less than $n$. Is there always an integer $r\geq 1$ such that $\binom{n}{r}$ is odd and divisible by $p$. Note that this is not true for the case $n=9$. Is it also true for $n$ even?

Thanks you very much in advance for comments and advice.

`http://en.wikipedia.org/wiki/Lucas'_theorem`

$\endgroup$ – Chris Godsil Jun 23 '12 at 15:23