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Combinatorics is the field in mathematics that deals with counting, arranging, and analysing the ways in which objects can be selected and organized. It is a fundamental skill in probability as it helps in determining the total possible outcomes of a scenario. In the urn problem, combinatorics is employed to calculate the total number of ways a ball can be drawn from each of three urns.

Each urn has a certain number of balls. To find total possible combinations of drawing one ball from each urn, you multiply the number of balls in Urn A, B, and C.

• Urn A has 9 balls (4 white and 5 blue)
• Urn B has 7 balls (4 white and 3 blue)
• Urn C has 6 balls (2 white and 4 blue)

Thus, the total number of outcomes is calculated as:\[9 \times 7 \times 6 = 378\]Understanding these basic principles of combinatorics can simplify the solving of more complex probability problems.

Conditional probability is a measure of the probability of an event occurring given that another event has already occurred. In many probability problems, you have to consider each event in the context of other events.

In this problem, we analyze the probabilities of drawing specific types of balls under certain conditions. For instance, if we want one blue and two white balls from the urns, different conditions arise based on which urn you pick a certain colored ball from.

• Case 1: White ball from A and B, Blue from C
• Case 2: White ball from A, Blue from B, White from C
• Case 3: Blue ball from A, White from B and C

Each case reflects a specific chain of events, showing how conditions influence probabilities, such as picking a white ball or a blue ball based on what's available in each urn. The conditional probabilities are applied to each scenario:\[\text{Probability for Case 1} = \frac{4}{9} \times \frac{4}{7} \times \frac{4}{6} = \frac{64}{378}\]Analyzing each condition helps in breaking down the problem into manageable parts, leading to a smoother calculation of overall probabilities.

Effective problem-solving in probability often involves systematic approaches to break down complex problems into simpler parts. In our urn example, strategic problem-solving techniques include dividing the problem into clear cases and calculating probabilities for each separately.

First, you identify all possible cases by organizing them based on specific outcomes. This entails recognizing that the desired result — two white balls and one blue ball — can occur in multiple distinct orders.

Second, calculate the probabilities for each case by identifying how each sequence can occur, like finding the probability of white coming from one urn, blue from another, and so on.

• Calculate probability for each case separately.
• Sum the probabilities of all favorable cases.
• Ensure to simplify fractions to find simplest form of the probability.

By concentrating on each scenario individually and ensuring each step is thoroughly calculated, you can reach a solution that is as simple as \(\frac{16}{47}\) in this particular problem. Problem-solving in probability is about being organized, patient, and methodical.