Considering the construction of a matrix as follows.
The $n$th row in the matrix is filled with the coeffcients of $x^r$ in the expansion of $(1+x)^n$ from the columns $2n$ to $3n$ inclusive and circle all the numbers that are divisible by $n$ in the same row
How would I find the number of columns for which all the elements in a column are circled in the first j columns given j = 547 ?
Hint: Have you tried looking at Pascal's triangle modulo some small primes? You could see a pattern that would apply. The Divisibility properties section of Wikipedia's Binomial Coefficient article has some useful information. This page has some neat images mod 2.