The sides of a triangle are 2006 cm, 6002 cm and 'm' cm, where m is a positive integer. How many values of 'm' are possible?

By the Triangle Inequality Theorem, the sum of any two sides of a triangle must be strictly greater than the third side.

The three sides are $2006$, $6002$, and $m$.

This gives us three conditions:

$2006 + 6002 > m \Rightarrow m < 8008$

$2006 + m > 6002 \Rightarrow m > 3996$

$6002 + m > 2006 \Rightarrow m > -3996$ (always true since $m$ is positive)

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Combining the useful conditions:

$3996 < m < 8008$

Since $m$ is a positive integer, $m$ can take values from $3997$ to $8007$.

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Total number of values $= 8007 - 3997 + 1 = 4011$