The question asks to find the probability that the sum of two numbers drawn from a set of balls numbered 1 through 10 is 9. The draws are without replacement.

Step-by-step solution:

1. Identify pairs of numbers that sum to 9: The possible pairs of numbers from 1 to 10 that add up to 9 are:

   • (1, 8)
   • (2, 7)
   • (3, 6)
   • (4, 5)

   Therefore, there are 4 pairs that satisfy the condition.

2. Calculate the total number of possible pairs: Since the question involves drawing 2 balls from a set of 10 without replacement, the total number of ways to choose 2 balls from 10 is given by the combination formula: $\binom{10}{2} = \frac{10 \times 9}{2} = 45$ So, there are 45 possible pairs in total.

3. Calculate the probability: The probability is the ratio of favorable outcomes (pairs that sum to 9) to the total number of possible outcomes (total pairs): $P = \frac{4}{45}$

Thus, the probability that the sum of the two numbers is 9 is $\frac{4}{45}$.

The correct answer is B) $\frac{4}{45}$.

Would you like further clarification or more details on any step? Here are some related questions:

1. What is the general formula for calculating combinations?
2. How does probability change when drawing with replacement versus without replacement?
3. How can you generalize this method for any sum from 2 to 20 when drawing from numbers 1 to 10?
4. How do you compute the probability of sums with more than two numbers drawn?
5. What are other scenarios where combinations are useful in probability?

Tip: When dealing with probabilities in card or number drawing problems, always start by identifying favorable outcomes and the total possible outcomes!