Multiple choice

A drawer contains $(12x + 3)$ red & $(5x + 1)$ black balls, when two balls are drawn at random, the probability that they both are red is $\dfrac{1}{2}$. The number of balls in drawer can be?

  1. $21$
  2. $11$
  3. $4$
  4. $3$
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A Correct answer
Explanation

Total balls = (12x + 3) + (5x + 1) = 17x + 4. Probability of 2 red balls = [(12x+3)/(17x+4)] * [(12x+2)/(17x+3)] = 1/2. Testing x=1 gives 15 red, 6 black, total 21. Probability = (15/21) * (14/20) = (5/7) * (7/10) = 35/70 = 1/2. Thus, total balls = 21.

AI explanation

The total number of balls in the drawer is the sum of red and black balls, which is 12x + 3 + 5x + 1, giving 17x + 4. The probability of drawing two red balls is the combination of 12x + 3 taken 2 at a time divided by the combination of 17x + 4 taken 2 at a time, which equals 1/2. This gives the equation 2 times (12x + 3) times (12x + 2) equals (17x + 4) times (17x + 3). Expanding both sides gives 288x squared plus 120x plus 12 equals 289x squared plus 119x plus 12, which simplifies to x squared minus x equals 0. The valid solution for x is 1. Substituting x equals 1 into the total balls formula 17x + 4 gives 17 plus 4, which is 21. The result is 21.

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