\\frac { \left| a \right| }{ a } \\) = 1 when a is positive. \\frac { \left| a \right| }{ a } \\) = -1 when a is negative.
\\ S = 1+\frac { \left| a \right| }{ a } +\frac {2 \left| b \right| }{ b } +\frac {3 \left| ab \right| }{ ab } -\frac {4 \left| c \right| }{ c } \\)
The value of the expression will be maximum when all of the terms become positive.
i.e., a, b, and ab should be positive and c should be negative.
When a is positive, \\frac { \left| a \right| }{ a } \\) = 1.
When b is positive, \\frac {2 \left| b \right| }{ b } \\) = 2.
When a and b are positive, as required in the previous two steps, ab will be positive and the expression, \\frac {3 \left| ab \right| }{ ab } \\) = 3.
When c is negative, \\frac {4 \left| c \right| }{ c } \\) = -4.
Therefore, the maximum value = 1 + 1 + 2 + 3 -(-4) = 11.
The value of the expression will be minimum if we make as many terms negative as possible. Higher the magnitude of the terms made negative, lower the value of the expression.
\\ S = 1+\frac { \left| a \right| }{ a } +\frac {2 \left| b \right| }{ b } +\frac {3 \left| ab \right| }{ ab } -\frac {4 \left| c \right| }{ c } \\)
c has to be positive for S to be minimum. The last term will then be -4.
If ab is negative, then \\frac {3 \left| ab \right| }{ ab } \\) = -3.
If ab has to be negative, one of a or b has to be positive and the other has to be negative.
Possibility 1: If a > 0 and b < 0, \\frac { \left| a \right| }{ a } \\) = 1 and \\frac {2 \left| b \right| }{ b } \\) = -2.
The value of the expression is 1 + 1 - 2 - 3 - 4 = -7.
Possibility 2: If a < 0 and b > 0, \\frac { \left| a \right| }{ a } \\) = -1 and \\frac {2 \left| b \right| }{ b } \\) = 2.
The value of the expression is 1 - 1 + 2 - 3 - 4 = -5.
Therefore, the minimum value of the expression is -7.
Maximum value = 11.
Minimum value = -7
The difference is 18.
Choice E is the correct answer.