Final answer:
The probability of selecting exactly 3 black buttons from a jar containing 6 black and 4 white buttons when 5 buttons are picked at random is calculated by determining the number of favorable combinations divided by the total number of possible combinations, resulting in a probability of 5/12.
Step-by-step explanation:
The question asks for the probability of selecting exactly 3 black buttons from a jar containing 6 black and 4 white buttons when picking 5 buttons at random. We are looking for the combination of 3 black and 2 white buttons in our selection.
To solve this, we need to calculate the number of favorable combinations and then divide by the total number of combinations possible for picking 5 buttons out of 10. The number of ways to choose 3 black buttons from a total of 6 is given by combinations, which is calculated by the formula C(n, k) = n! / (k!(n-k)!), where n is the total number of items to pick from, k is the number of items to pick, and '!' denotes factorial.
The number of combinations for picking 3 black buttons is C(6, 3), and the number of combinations for picking 2 white buttons from the remaining 4 is C(4, 2). Thus, the number of favorable outcomes is C(6, 3) × C(4, 2). The total number of ways to pick any 5 buttons from the 10 is C(10, 5).
The probability is then the number of favorable outcomes divided by the total number of outcomes: P(3 black) = [C(6, 3) × C(4, 2)] / C(10, 5).
Calculating this gives us P(3 black) = [20 × 6] / 252 = 120 / 252 = 5/12. So, the correct answer is d) 5/12