Let's go through question (c) step by step.

Given:

• Set $A = \{x : 10 \leq x \leq 21, x \text{ is an integer}\}$.
• We need to find the probability of two specific events if a number is chosen randomly from set $A$:

1. Event (i): The number chosen is a multiple of 3.
2. Event (ii): The sum of the two digits of the number is even.

Step 1: Identify the Elements of Set $A$

The integers between 10 and 21 are: $A = \{10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21\}$ There are 12 elements in total.

Step 2: Find Probability for Event (i)

We are looking for numbers in $A$ that are multiples of 3. The multiples of 3 within this range are: $12, 15, 18, 21$ So, there are 4 numbers that meet this criterion.

The probability of Event (i) is: $P(\text{multiple of 3}) = \frac{4}{12} = \frac{1}{3}$

Step 3: Find Probability for Event (ii)

For Event (ii), we need the sum of the two digits to be even. Let's calculate the digit sums for each element in $A$:

• $10: 1 + 0 = 1$ (odd)
• $11: 1 + 1 = 2$ (even)
• $12: 1 + 2 = 3$ (odd)
• $13: 1 + 3 = 4$ (even)
• $14: 1 + 4 = 5$ (odd)
• $15: 1 + 5 = 6$ (even)
• $16: 1 + 6 = 7$ (odd)
• $17: 1 + 7 = 8$ (even)
• $18: 1 + 8 = 9$ (odd)
• $19: 1 + 9 = 10$ (even)
• $20: 2 + 0 = 2$ (even)
• $21: 2 + 1 = 3$ (odd)

The numbers with an even digit sum are: $11, 13, 15, 17, 19, 20$. There are 6 such numbers.

The probability of Event (ii) is: $P(\text{even digit sum}) = \frac{6}{12} = \frac{1}{2}$

Final Answers

1. Probability that the number is a multiple of 3: $\frac{1}{3}$.
2. Probability that the sum of the two digits is even: $\frac{1}{2}$.

Would you like more details on any of these steps or have additional questions?

Here are some related questions to explore further:

1. What is the probability of choosing a number that is both a multiple of 3 and has an even digit sum?
2. How would the probabilities change if the range of $x$ extended from 10 to 30?
3. Can we determine the probabilities for numbers that are multiples of other values (like 4 or 5)?
4. What is the probability of selecting a prime number from set $A$?
5. If we chose two numbers from set $A$, what is the probability both are even?

Tip: When calculating probabilities, always ensure you understand the total number of possible outcomes to set the correct denominator.