swerve wrote:A jar contains 12 marbles consisting of an equal number of red, green and blue marbles. Four marbles are removed from the jar and discarded. What is the probability that only two colors will remain in the jar after the four marbles have been removed?

A. 1/495
B. 1/165
C. 1/81
D. 1/3
E. 1/2

GMATH´s method suggests avoiding the "partial probabilities sequences" (correctly used in previous post) because:

1. The justification for its validity is out-of-GMAT´s scope (conditional probabilities, NOT independency).
2. There are some problems in which students usually fall into "traps".

All that mentioned, let´s see how we would deal with this problem!

\[4\,{\text{red}}\,\,,\,\,4\,\,{\text{green,}}\,\,{\text{4}}\,\,{\text{blue}}\,\,\,\, \to \,\,\,\,{\text{4}}\,\,{\text{taken}}\,\,{\text{out}}\]
\[? = P\left( {{\text{only}}\,\,{\text{2}}\,\,colors\,\,{\text{remain}}} \right)\,\,\,\, \Leftrightarrow \,\,\,\,\boxed{? = P\left( {4\,\,{\text{taken}}\,\,{\text{out}}\,\,{\text{same}}\,\,{\text{color}}} \right)}\]
\[{\text{total}} = C\left( {12,4} \right)\,\,\,\,\left( {{\text{equiprobables}}} \right)\]
\[{\text{favorable}}\,\,{\text{ = }}\,\,3\,\,\,\,\,\,\left( {{\text{all}}\,\,{\text{red}}\,\,{\text{or}}\,\,{\text{all}}\,\,{\text{green}}\,\,{\text{or}}\,\,{\text{all}}\,\,{\text{blue}}} \right)\]
\[? = \frac{3}{{C\left( {12,4} \right)}} = \frac{{3\, \cdot 4!}}{{12 \cdot 11 \cdot 10 \cdot 9}} = \ldots = \frac{1}{{165}}\]

The above follows the notations and rationale taught in the GMATH method.