1. Key Concepts and Formulas

• Probability of an Event: The probability $P(E)$ of an event $E$ is defined as the ratio of the number of favorable outcomes for $E$, denoted $n(E)$, to the total number of possible outcomes in the sample space $S$, denoted $n(S)$. $P(E) = \frac{n(E)}{n(S)}$
• Combinations: When selecting $k$ distinct items from a set of $n$ distinct items, where the order of selection does not matter, the number of ways is given by the combination formula: $^n C_k = \frac{n!}{k!(n-k)!}$
• Modular Arithmetic: This mathematical system deals with remainders after division. The notation $a \equiv b \pmod m$ means that $a$ and $b$ have the same remainder when divided by $m$, or equivalently, $a-b$ is a multiple of $m$. Key properties include:
  • If $a \equiv b \pmod m$ and $c \equiv d \pmod m$, then $a+c \equiv b+d \pmod m$ and $a-c \equiv b-d \pmod m$.
  • $N$ is a multiple of $m$ if and only if $N \equiv 0 \pmod m$.

2. Step-by-Step Solution

Step 1: Determine the Total Number of Possible Outcomes ($n(S)$)

First, we need to find the total number of ways to select two different numbers from the given set.

• Identify the Set: The given set is $A = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$. The number of elements in this set is $11$.
• Understand Selection Criteria: We are choosing "two different numbers." This implies:
  • The numbers must be distinct (e.g., cannot choose 5 and 5).
  • The order of selection does not matter (e.g., choosing (1, 2) is the same as choosing (2, 1) because the conditions apply to the pair, not to a specific first or second number).
• Calculate using Combinations: Since order doesn't matter and numbers are distinct, we use the combination formula $^n C_k$ with $n=11$ (total numbers) and $k=2$ (numbers to choose). $n(S) = ^{11}C_2 = \frac{11!}{2!(11-2)!} = \frac{11!}{2!9!}$ $n(S) = \frac{11 \times 10 \times 9!}{2 \times 1 \times 9!} = \frac{11 \times 10}{2} = 11 \times 5 = 55$ There are $55$ distinct ways to choose two different numbers.

Step 2: Simplify Conditions using Modular Arithmetic

Let the two chosen numbers be $x$ and $y$. The problem states two conditions:

1. Their sum is a multiple of 4: $x+y \equiv 0 \pmod 4$.
2. Their absolute difference is a multiple of 4: $|x-y| \equiv 0 \pmod 4$.

Since $k \equiv 0 \pmod 4$ implies $|k| \equiv 0 \pmod 4$, and vice-versa (if $|k|$ is a multiple of 4, then $k$ must also be a multiple of 4), the second condition can be simplified to $x-y \equiv 0 \pmod 4$. (The choice of $x-y$ or $y-x$ doesn't change the modular property).

We now have a system of congruences:

1. $x+y \equiv 0 \pmod 4$
2. $x-y \equiv 0 \pmod 4$

Let's deduce properties of $x$ and $y$:

• Adding the two congruences: $(x+y) + (x-y) \equiv 0+0 \pmod 4$ $2x \equiv 0 \pmod 4$ This means $2x$ must be a multiple of 4. For $2x$ to be a multiple of 4, $x$ itself must be an even number. If $x$ were odd, say $x=2k+1$, then $2x=2(2k+1)=4k+2$, which is never a multiple of 4 (it leaves a remainder of 2). Therefore, $x$ must be even.

• Subtracting the second congruence from the first: $(x+y) - (x-y) \equiv 0-0 \pmod 4$ $2y \equiv 0 \pmod 4$ Following the same logic as for $x$, $2y$ must be a multiple of 4, which implies $y$ must also be an even number.

Crucial Deduction: Both numbers $x$ and $y$ must be even to satisfy both conditions simultaneously. This significantly narrows down our search for favorable outcomes.

Step 3: Identify and Enumerate Favorable Outcomes ($n(E)$)

From the original set $A = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$, let's list all the even numbers: $E_{numbers} = \{0, 2, 4, 6, 8, 10\}$.

Now, we need to select two different numbers from $E_{numbers}$ such that their sum and difference are both multiples of 4. We will systematically list pairs $(x, y)$ where $x < y$ to avoid duplicates.

• Starting with 0:

  • $(0, 2)$: Sum = 2 (not mult of 4), Diff = 2 (not mult of 4)
  • $(0, 4)$: Sum = 4 (mult of 4), Diff = 4 (mult of 4) $\rightarrow$ Favorable: (0, 4)
  • $(0, 6)$: Sum = 6 (not mult of 4), Diff = 6 (not mult of 4)
  • $(0, 8)$: Sum = 8 (mult of 4), Diff = 8 (mult of 4) $\rightarrow$ Favorable: (0, 8)
  • $(0, 10)$: Sum = 10 (not mult of 4), Diff = 10 (not mult of 4)

• Starting with 2: (only consider numbers greater than 2)

  • $(2, 4)$: Sum = 6 (not mult of 4), Diff = 2 (not mult of 4)
  • $(2, 6)$: Sum = 8 (mult of 4), Diff = 4 (mult of 4) $\rightarrow$ Favorable: (2, 6)
  • $(2, 8)$: Sum = 10 (not mult of 4), Diff = 6 (not mult of 4)
  • $(2, 10)$: Sum = 12 (mult of 4), Diff = 8 (mult of 4) $\rightarrow$ Favorable: (2, 10)

• Starting with 4: (only consider numbers greater than 4)

  • $(4, 6)$: Sum = 10 (not mult of 4), Diff = 2 (not mult of 4)
  • $(4, 8)$: Sum = 12 (mult of 4), Diff = 4 (mult of 4) $\rightarrow$ Favorable: (4, 8)
  • $(4, 10)$: Sum = 14 (not mult of 4), Diff = 6 (not mult of 4)

• Starting with 6: (only consider numbers greater than 6)

  • $(6, 8)$: Sum = 14 (not mult of 4), Diff = 2 (not mult of 4)
  • $(6, 10)$: Sum = 16 (mult of 4), Diff = 4 (mult of 4) $\rightarrow$ Favorable: (6, 10)

• Starting with 8: (only consider numbers greater than 8)

  • $(8, 10)$: Sum = 18 (not mult of 4), Diff = 2 (not mult of 4)

The list of favorable pairs is: $\{(0, 4), (0, 8), (2, 6), (2, 10), (4, 8), (6, 10)\}$ Counting these, we find that the number of favorable outcomes, $n(E)$, is $6$.

Step 4: Calculate the Probability

Using the values found:

• Number of favorable outcomes, $n(E) = 6$
• Total number of possible outcomes, $n(S) = 55$

The probability $P(E)$ is: $P(E) = \frac{n(E)}{n(S)} = \frac{6}{55}$

3. Common Mistakes & Tips

• Misinterpreting "Different Numbers": Always ensure you're selecting distinct numbers. If the order doesn't matter (as in this case for sum/difference), use combinations. If order matters (e.g., forming a sequence), use permutations.
• Neglecting Modular Arithmetic: Directly checking all 55 pairs for the conditions would be highly time-consuming and prone to errors. Using modular arithmetic to deduce that both numbers must be even is a crucial shortcut.
• Incomplete/Duplicated Listing: When enumerating favorable outcomes, adopt a systematic approach (e.g., increasing order for the first number, then increasing order for the second) to ensure all unique favorable pairs are counted exactly once.

4. Summary

To solve this probability problem, we first calculated the total number of ways to select two distinct numbers from the given set using combinations. Then, we applied modular arithmetic to simplify the conditions for favorable outcomes, deducing that both selected numbers must be even. Finally, we systematically listed and counted the pairs of even numbers that satisfied both the sum and absolute difference conditions, leading to the number of favorable outcomes. The probability was then calculated as the ratio of favorable to total outcomes.

The final answer is $\boxed{\frac{6}{55}}$, which corresponds to option (D).