Permutations and Combinations
If there are 5 dishes, they can be relished in any order at a time. In permutation, it should be in a particular order. In combination, the order does not matter. Take 3 letters a, b, and c. The possible ways of pairing any two letters are ab, bc, ac, ba, cb and ca. It is in a particular order. So, this can be called the permutation of a, b, and c. But if the order does not matter then ab is the same as ba. Similarly, bc is the same as cb and ac is the same as ca. Here the list has ab, bc, and ac alone. This can be called the combination of a, b, and c.
Counting Theory
The fundamental counting principle is a rule that is used to count the total number of possible outcomes in a given situation.
![### Geometry Problem: Understanding Rectangles and Properties of Angles
#### Given:
- **Rectangle ABCD**
- \(\angle BDC = 2x - 40\)
- \(\angle BDA = 30\)
#### Tasks:
1. **Copy and Label** the diagram.
2. **Solve for \(x\)**.
3. **What property allows you to set up and solve for \(x\)?**
#### Diagram:
The diagram represents a rectangle \(ABCD\) with diagonals \(AC\) and \(BD\) intersecting at point \(E\). Each corner of the rectangle (\(A\), \(B\), \(C\), \(D\)) and the intersection point \(E\) is labeled accordingly.
---
To solve the problem:
1. **Copy and Label**:
- Draw a rectangle and name the vertices \(A\), \(B\), \(C\), and \(D\) in a clockwise or counterclockwise order.
- Draw the diagonals \(AC\) and \(BD\) intersecting at point \(E\).
- Label angles \(\angle BDC\) and \(\angle BDA\) as given: \(\angle BDC = 2x - 40\) and \(\angle BDA = 30\).
2. **Solve for \(x\)**:
- According to the property of rectangles, diagonals bisect each other. Hence, \(\angle BDA = \angle BDC\).
- Set the expressions equal to each other: \(2x - 40 = 30\).
- Solve the equation:
\[
2x - 40 = 30
\]
\[
2x = 70
\]
\[
x = 35
\]
3. **Property Used**:
- The property used here is the fact that in a rectangle, the diagonals are congruent and bisect each other. This makes the angles formed by these diagonals congruent.
By following these steps, you can understand how to label diagrams correctly, set up and solve equations based on geometric properties, and apply these concepts to solve for unknown variables.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8afafbd6-82ff-4c2b-b847-f943418ec019%2Fb4af11e3-bcfe-4623-aea3-d4389d7baf81%2Fweh9ixq_processed.jpeg&w=3840&q=75)
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