Step-by-step Solution:

Correct Answer: C (10)

The problem asks for the number of elements in a relation R. An element of R is an ordered pair $(x, y)$ that satisfies three conditions simultaneously.

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Step 1: Understand the Conditions for the Relation R

An ordered pair $(x, y)$ is an element of the relation R if it meets the following criteria:

1. $x$ must be an element of set $A = \{1, 2, 3, \dots, 20\}$. This means $x$ is an integer and $1 \le x \le 20$.
2. $y$ must also be an element of set $A$. This means $y$ is an integer and $1 \le y \le 20$.
3. The pair must satisfy the rule $y = 2x - 7$.

To find the number of elements in R, we need to find how many integer values of $x$ from 1 to 20 produce an integer value of $y$ that is also between 1 and 20.

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Step 2: Set Up and Solve the Inequalities

We use the constraint on $y$ (condition 2) and substitute the rule for the relation (condition 3) into it.

$$ 1 \le y \le 20 $$

Substitute $y = 2x - 7$:

$$ 1 \le 2x - 7 \le 20 $$

We can solve this compound inequality for $x$ by adding 7 to all parts:

$$ 1+7 \le 2x \le 20+7 $$

$$ 8 \le 2x \le 27 $$

Now, divide all parts by 2:

$$ 4 \le x \le 13.5 $$

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Step 3: Count the Number of Valid Elements

We have found the range for $x$. Now we must also apply the first condition: $x$ must be an integer from the set A.

We need to count the number of integers $x$ such that $4 \le x \le 13.5$.

The possible integer values for $x$ are: 4, 5, 6, 7, 8, 9, 10, 11, 12, and 13.

By counting these values, we find that there are 10 possible values for $x$. Each of these values will produce a unique corresponding $y$ value that is within the set A, thus forming a valid element in R.

Therefore, the number of elements in R is 10.