Coordinates Of A Point In Space

The formula for the number of permutations of objects taken from a set of objects is given by: In the given question, represents the total number of objects, and represents the number of objects chosen. So: This is equivalent to: Thus, the correct answer is .

Step 1: Use the formula for the slope of a line.
The points (3, −4) and (4, −a) lie on the same straight line. The slope of a line through two points and is given by the formula: Substitute the given points (3, −4) and (4, −a) into the formula:
Step 2: Use the condition for points on the same line.
Since the points lie on the same line, their slope must be the same. Now, calculate the slope using the coordinates of the first point (3, −4) and the second point (4, −a).

Step 1: Use the principle of inclusion-exclusion.
We are given the following information: - - From the principle of inclusion-exclusion, we know that: Substituting the given values:
Step 2: Find .
The complement of is denoted as , and by the complement rule, we know: where is the total number of elements in the universal set. Since and this represents all the elements in the universal set, we have . Therefore: