Number of solutions

(1+1a)(1+1b)(1+1c)(1+1d)(1+1e)=14.4 \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)(a, b, c, d, e) that satisfy the equation above.

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