A bag contains 20 blue marbles, 12 red marbles and some other number of green marbles. If the probability of drawing green marble in one try is $\frac{1}{y}$ then which of the following statements is/are correct?
Number of possible integral values of $y$ is 5
Number of possible integral values of $y$ is 6
Sum of possible integral values of $y$ is 69
Sum of possible integral values of $y$ is 96
Step-by-Step Solution
Key Concept: Given P(green) = 1/y = x/(x+32) where x is the number of green marbles, cross-multiplying yields x = 32/(y-1). Since x must be a positive integer, y-1 must be a divisor of 32, constraining y to specific values that produce integer solutions.
Let $x$ be the number of green marbles. The probability of drawing a green marble in one try is $\frac{1}{y} = \frac{x}{x+32}$. Cross-multiplying: $x + 32 = yx$, so $x = \frac{32}{y-1}$. Since $x$ must be a positive integer, possible values of $y$ are $\{33, 17, 9, 5, 3, 2\}$ with corresponding values of $x$ being $\{1, 2, 4, 8, 16, 32\}$.
Correct Answer: 2,3