Question

A box contains 8 red and $x$ number of green balls. 3 balls are drawn at random, if the probability that 3 balls being red is $\frac{7}{15}$, then number of green balls is ______.

• A. 2
• B. 4
• C. 3
• D. 5

Hint:

Never expand $(x+8)(x+7)(x+6)$ into a cubic equation! Just find the prime factorization of 720 and group the numbers into three consecutive integers.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given the probability of drawing 3 red balls sequentially from a mixed box. We need to work backwards using combinatorial probability to find the unknown number of green balls ($x$).

Step 2: Detailed Explanation:
Number of Red balls = 8
Number of Green balls = $x$
Total number of balls = $8 + x$
The total number of ways to draw any 3 balls from the box is the sample space: $^{(8+x)}\text{C}_3$.
The number of ways to successfully draw 3 Red balls from the 8 available is: $^8\text{C}_3$.
The probability of drawing 3 Red balls is:
$P(\text{3 Red}) = \frac{^8\text{C}_3}{^{(8+x)}\text{C}_3}$
We are given $P(\text{3 Red}) = \frac{7}{15}$. Equating them:
$\frac{^8\text{C}_3}{^{(8+x)}\text{C}_3} = \frac{7}{15}$
Expand the combination formulas ($^n\text{C}_r = \frac{n!}{r!(n-r)!}$):
$^8\text{C}_3 = \frac{8 \times 7 \times 6}{3 \times 2 \times 1} = 56$
$^{(8+x)}\text{C}_3 = \frac{(x+8)(x+7)(x+6)}{3 \times 2 \times 1}$
Substitute these back into the probability equation:
$\frac{56}{\frac{(x+8)(x+7)(x+6)}{6}} = \frac{7}{15}$
Multiply numerator by 6:
$\frac{336}{(x+8)(x+7)(x+6)} = \frac{7}{15}$
Cross-multiply to solve for the polynomial:
$7 \times (x+8)(x+7)(x+6) = 336 \times 15$
$(x+8)(x+7)(x+6) = \frac{5040}{7}$
$(x+8)(x+7)(x+6) = 720$
We are looking for three consecutive descending integers that multiply to 720.
Notice that $10 \times 9 \times 8 = 720$.
Comparing terms:
$x + 8 = 10 \implies x = 2$
$x + 7 = 9 \implies x = 2$
$x + 6 = 8 \implies x = 2$
The number of green balls is 2.

Step 3: Final Answer:
The number of green balls is 2, matching option (a).