There are $5$ yellow pegs, $4$ red pegs, $3$ green pegs, $2$ blue pegs, and $1$ orange peg to be placed on a triangular peg board. Find the number of ways the pegs can be placed so that no (horizontal) row or (vertical) column contains two pegs of the same color.

Let's think step by step and output the final answer within \boxed{}.
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