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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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    Is the answer `547C2` where nCr = n!/(r!*(n-r)!) ?2011-01-28
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    Can you please provide some more insight?2011-01-28
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    It can't be that big. You only have 547 columns to consider. Start with 3's. In row 3, columns 6 through 9 have 1, 3, 3, and 1, and you circle the two 3's in 7 and 8. Row 4 has 1,4,6,4,1 in columns 8 through 12 and you circle the 4's in 9 and 11. So column 8 doesn't have all the numbers circled, but 7 does. If you make a spreadsheet going up to 20 or so you will probably see the pattern.2011-01-28
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    Could I get an explanation for the downvote?2011-01-28
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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