A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is: [2026]

Question

A box contains 5 blue, 6 yellow and 4 red balls. The number of ways, of drawing 8 balls containing at least two balls of each colour, is: [2026]

Smart Achievers · Accepted Answer

(1)

We first select 6 balls (2 Blue, 2 Yellow, 2 Red). Now we only need to select 2 more balls from the remaining balls.

For each colour, the coefficient of 'x' represents the number of ways to select and additional balls:

Case I: Blue (5 total): We already took 2, so 3 remaining.

Number of ways to select (0,1, or 2):

$(\begin{matrix} 4 \\ 2 \end{matrix}) + (\begin{matrix} 4 \\ 3 \end{matrix}) x + (\begin{matrix} 4 \\ 4 \end{matrix}) x^{2} = 6 + 4 x + x^{2}$

Case II: Yellow (6 total): We already took 2, or 4 remaining.

Number of ways to select (0, 1, or 2):

$(\begin{matrix} 6 \\ 2 \end{matrix}) + (\begin{matrix} 6 \\ 3 \end{matrix}) x + (\begin{matrix} 6 \\ 4 \end{matrix}) x^{2} = 15 + 20 x + 15 x^{2}$

Case III: Red (4 balls): We already took2, so 2 remaining.

Number of ways to pick (0, 1, or 2):

$(\begin{matrix} 4 \\ 2 \end{matrix}) + (\begin{matrix} 4 \\ 3 \end{matrix}) x + (\begin{matrix} 4 \\ 4 \end{matrix}) x^{2} = 6 + 4 x + x^{2}$

To get 2 more balls, we look for combinations of terms from each bracket where the powers of x add up to 2:

Required number of ways = 450 + 900 + 150 + 1200 + 600 + 800 = 4100

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