Find the remainder

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A soccer team has $22$ available players. A fixed set of $11$ players starts the game, while the other $11$ are available as substitutes. During the game, the coach may make as many as $3$ substitutions, where any one of the $11$ players in the game is replaced by one of the substitutes. No player removed from the game may reenter the game, although a substitute entering the game may be replaced later. No two substitutions can happen at the same time. The players involved and the order of the substitutions matter. Let $n$ be the number of ways the coach can make substitutions during the game ,including the possibility of making no substitutions. Find the remainder when $n$ is divided by $1000$

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There are d66eb577e8abc465504cef8ddd0370390b6962ff substitutions. The number of ways to sub any number of times must be multiplied by the previous number. This is defined recursively. The case for bc1f9d9bf8a1b606a4188b5ce9a2af1809e27a89 subs is dce34f4dfb2406144304ad0d6106c5382ddd1446, and the ways to reorganize after 174fadd07fd54c9afe288e96558c92e0c1da733a subs is the product of the number of new subs (bb270cb315a1e19d3e6f0dc421c43acd8b920b64) and the players that can be ejected (c6878713578626763c38433b3f4c8c2205ad0c15). The formula for 174fadd07fd54c9afe288e96558c92e0c1da733a subs is then e64561bf0e25c82c0aa4c6eb55e81f839be32b79 with a16f2d4ee29c282b705559743af4c40867c16c88.

Summing from bc1f9d9bf8a1b606a4188b5ce9a2af1809e27a89 to 7cde695f2e4542fd01f860a89189f47a27143b66 gives c241fb12f3502957d2380eb7a43c33a9527e4a88. Notice that 712e0d3f79b62393937d5d7d7677d0114d2f2876. Then, rearrange it into 5e86bee9086d70a8dd37921e7791712620e078ef. When taking modulo 6ee927e1332358c96c62c277441c907c4f51057f, the last term goes away. What is left is d76819764930adb714755efed97a02ebd6422500

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