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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 ?

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    "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 ?" It feels that one of the "_columns_" must actually read "_rows_". Is the question correct?2011-07-27

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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.

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    Also, any row that is a multiple of 2 or 3 will have a 1 in it (like row 3 in my example), which you won't circle. So you only have to consider rows of the form 6k+1 and 6k+5.2011-01-28