\[ \left( 1+\dfrac1a\right)\left( 1+\dfrac1b\right)\left( 1+\dfrac1c\right)\left( 1+\dfrac1d\right)\left( 1+\dfrac1e\right) = 14.4 \]

Find the number of 5-tuples of positive integers \((a, b, c, d, e)\) that satisfy the equation above.

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