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  "body": "\n## Summary\n\nThe falling factorial $x^{\\underline{n}}$ multiplies $n$ consecutive factors going down from $x$, and the rising factorial $x^{\\overline{n}}$ multiplies $n$ consecutive factors going up from $x$.\nBoth are polynomials of degree $n$ in $x$, and both reduce to the [[factorial|0XWG0JGDFW]] $n!$ for a suitable choice of $x$.\nThe falling factorial counts ordered selections without repetition, and the rising factorial (the Pochhammer symbol of special functions) appears in hypergeometric series.\nThe node [[falling factorial|0SJNCJCCQG]] treats the falling case alone.\n\n## Explanations\n\n### L1 human\n\nStart at a number and multiply it by the next few numbers below it: that is a falling factorial.\nMultiply it by the next few numbers above it instead: that is a rising factorial.\nFor example, $5^{\\underline{3}} = 5 \\cdot 4 \\cdot 3 = 60$ and $5^{\\overline{3}} = 5 \\cdot 6 \\cdot 7 = 210$.\nThe number $60$ counts the ways to give gold, silver and bronze medals to $3$ of $5$ runners.\nBoth products start at $x$ and use $n$ factors, so with $x = n$ falling or $x = 1$ rising they give $n!$.\n",
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