Let’s go through each question in the image to break them down and solve as needed:

Question 1

Given: A class has 100 students, with 40 males.
You’re asked different scenarios of forming a committee of 10 students.

1. (a) How many ways to form a 10-person committee?

   • Since there are no gender restrictions, the answer is simply the combination of 100 students taken 10 at a time: $C(100, 10)$.

2. (b) If the number of males must equal the number of females (5 males and 5 females)?

   • Here, select 5 males from 40, and 5 females from 60. The answer is the product of these two combinations: $C(40, 5) \times C(60, 5)$.

3. (c) If the committee must have either 6 males and 4 females or 4 males and 6 females?

   • For the first option (6 males, 4 females): $C(40, 6) \times C(60, 4)$.
   • For the second option (4 males, 6 females): $C(40, 4) \times C(60, 6)$.
   • The answer is the sum of these two results.

Question 2

Show that in any set of 6 classes, two will have the same day scheduled (assuming no classes on Saturday).

This is a Pigeonhole Principle problem. Since there are 5 weekdays and 6 classes, at least two classes will fall on the same day, as we have more classes than available days.

Question 3

What is the probability of randomly selecting an even number from the first 100 positive integers?

• The even numbers from 1 to 100 are {2, 4, ..., 100}, totaling 50 numbers.
• Probability = $\frac{50}{100} = 0.5$.

Question 4

Given the price data for rice of different qualities over several months, you are asked to:

1. (a) Find the average price deviation for each quality (Premium, Medium, Low).

   • Calculate the mean of the prices for each quality, then find the average deviation from this mean for each month.

2. (b) Find the standard deviation (SD) for each quality.

   • Calculate the standard deviation for the prices of each quality over the months provided.

Would you like to go deeper into any of these calculations, or should we start with specific parts?

Further Exploratory Questions:

1. How can the combination formulas be calculated in detail for each part of question 1?
2. What is the exact computation of probabilities using different sets in combinatorics?
3. Can the Pigeonhole Principle apply to more complex scheduling scenarios?
4. How does the mean deviation differ from standard deviation conceptually and computationally?
5. What are the statistical implications of price variations in economic analyses?

Tip:

For calculating combinations $C(n, k)$, use the formula $C(n, k) = \frac{n!}{k!(n-k)!}$.