{"status": "success", "data": {"description_md": "Define the *partition sum* of a positive integer as a collection of not necessarily distinct integers between $1$ and $3$, inclusive, such that their sum equals the given positive integer. For instance, one possible partition sum of $14$ is $\\{1, 3, 3, 2, 2, 2, 1\\},$ since $1+3+3+2+2+2+1 = 14$. Two partition sums are considered identical if they contain the same amount of $1$s, the same amount of $2$s, and the same amount of $3$s, in an order that is not necessarily the same. As an example, $\\{1, 2,2, 3\\}$ is equivalent to $\\{2,3,1,2\\}$, as both contain one $1$, two $2$s, and one $3$. \n\nDetermine the total number of unique partition sums for the number $15$.\n\n$(\\textbf{A})\\;25\\quad(\\textbf{B})\\;26\\quad(\\textbf{C})\\;27\\quad(\\textbf{D})\\;28\\quad(\\textbf{E})\\;29$", "description_html": "

Define the partition sum of a positive integer as a collection of not necessarily distinct integers between 1 and 3, inclusive, such that their sum equals the given positive integer. For instance, one possible partition sum of 14 is \\{1, 3, 3, 2, 2, 2, 1\\}, since 1+3+3+2+2+2+1 = 14. Two partition sums are considered identical if they contain the same amount of 1s, the same amount of 2s, and the same amount of 3s, in an order that is not necessarily the same. As an example, \\{1, 2,2, 3\\} is equivalent to \\{2,3,1,2\\}, as both contain one 1, two 2s, and one 3.

Determine the total number of unique partition sums for the number 15.

(\\textbf{A})\\;25\\quad(\\textbf{B})\\;26\\quad(\\textbf{C})\\;27\\quad(\\textbf{D})\\;28\\quad(\\textbf{E})\\;29

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