Finding number of integral solutions I am really getting confused in this question.
Number of integral solutions of the equation.
$x_1x_2x_3x_4=770$
options-


*

*$2^{11}$

*$2^{10}$

*$4^4$

*$5^5$
I attemtemted it by saying that $7*11*5*2=770$
so solutions should be $4!$
then we can say $77*5*2*1=770$
so $4!$
and $77*10*1*1=770$
so $4!/2$
and then $770*1*1*1=770$
so $4!/3!$
now multiplying them i get $27648$.
i am sure i have made a big mistage here but i dont know what.
please help

Case 1:  $2\cdot 5\cdot 7\cdot 11 = 770 \to 4! = 24$
Case 2: $10\cdot 7\cdot 11\cdot 1 = 770 \to 4!$
Case 3: $14\cdot 5 \cdot 11\cdot 1 = 770 \to 4!$
Case 4: $22 \cdot 5\cdot 7\cdot 1 = 770 \to 4!$
Case 5: $35 \cdot 2\cdot 11\cdot 1 = 770 \to 4!$
Case 6: $55 \cdot 2\cdot 7\cdot 1 = 770 \to 4!$
Case 7: $77\cdot 2\cdot 5\cdot 1 = 770 \to 4!$
Case 8: $70\cdot 11\cdot 1\cdot 1 = 770 \to 4\cdot 3 = 12$
Case 9: $110\cdot 7\cdot 1\cdot 1 = 770 \to 4\cdot 3 = 12$
Case 10: $154\cdot 5\cdot 1\cdot 1 = 770 \to 4\cdot 3 = 12$
Case 11: $385\cdot 2\cdot 1\cdot 1 = 770 \to 4\cdot 3 = 12$
Case 12: $770\cdot 1\cdot 1\cdot 1 = 770 \to 4$.
Thus the total number of solutions is: $7\cdot 24 + 4\cdot 12 + 4 = 220$
