2026 AMC 10A Problem 16

An urn contains $6$ red balls, $4$ black balls, and $4$ white balls. Suppose $7$ balls are drawn at random without replacement and a score is computed as follows:

$\cdot$ $1$ point is given for each red ball up to a maximum of $3$ points.

$\cdot$ $1$ point is given for each black ball up to a maximum of $2$ points.

$\cdot$ $1$ point is given for each white ball up to a maximum of $2$ points.

For example, drawing $1$ red ball, $4$ black balls, and $2$ white balls would earn a score of $1+2+2=5$ . What is the probability of getting the lowest possible score?