Factorial Countdown: Making numbers out of smaller ones

Let us play Factorial Countdown, a game in which you are given numbers $1,2,\ldots,n$, a set of arithmetic operations and a target number $N$. The aim of the game if to use all of the numbers, and some of the operations, to reach your target $N$. Can you win this game? This article describes when you can and when you cannot.

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