AMC 10A/12A 2026 (Problem 16/11)

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:

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

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

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?

\text{(A)}\;\frac{1}{858}

\text{(B)}\;\frac{5}{3432}

\text{(C)}\;\frac{1}{312}

\text{(D)}\;\frac{2}{429}

\text{(E)}\;\frac{5}{264}