# How many permutations with this set of rows and columns?

I have a range of possible page design layouts consisting of rows and columns in those rows. There are a maximum of 6 possible rows, and a maximum of 5 possible columns per row.

How many unique permutations of rows and columns are there, presuming that the columns are always equally spaced, so that there are only 5 column options per row?

Here are two examples of permutations:

Row 1: 4 columns
Row 2: 3 columns
Row 3: 5 columns

Row 1: 1 column
Row 2: 2 columns
Row 3: 2 columns
Row 4: 2 columns
Row 5: 2 columns
Row 6: 2 columns


What would be doubly helpful would be pseudocode for printing out all the variations! Thanks!

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I'm unsure what you mean by "the columns are always equally spaced".

If each row can have any number of columns, up to five, then perhaps the simplest thing to do is to find the number of layouts for a specific number of rows; then sum these quantities.

If there are $i$ rows, then there are $5^i$ different layouts.

For example, if there are two rows, then there are 5 choices for the number of columns in row 1 and 5 choices for the number of columns in row two; so there are $5^2$ layouts when you have two rows.

So, the total number of layouts is $$5 + 5^2+ 5^3+ 5^4+ 5^5+ 5^6.$$

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Thank you David. My comment about the columns being equally spaced just confused the issue - you answered the question as it was intended. –  Lemmy Jan 25 '12 at 17:00