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.
![### Problem 12: Right Triangle Analysis
A right triangle is presented in the diagram. The triangle has one angle marked as 30 degrees. The side opposite the right angle (hypotenuse) is labeled as 9 units. Let’s denote the sides opposite the 30-degree angle, the 60-degree angle, and the right angle as \(x\), \(y\), and 9 units respectively.
**Key Information:**
- The angle marked is \(30^\circ\).
- The hypotenuse is labeled as 9 units.
- The side opposite the 30-degree angle is labeled \(x\).
- The side adjacent to the 30-degree angle is labeled \(y\).
**Using Trigonometric Ratios:**
1. **Sine Function:**
\[
\sin(30^\circ) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{x}{9}
\]
Since \(\sin(30^\circ) = \frac{1}{2}\):
\[
\frac{1}{2} = \frac{x}{9} \implies x = \frac{9}{2} = 4.5
\]
2. **Cosine Function:**
\[
\cos(30^\circ) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{y}{9}
\]
Since \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\):
\[
\frac{\sqrt{3}}{2} = \frac{y}{9} \implies y = 9 \cdot \frac{\sqrt{3}}{2} = 4.5\sqrt{3}
\]
**Conclusion:**
- The side \(x\) opposite the 30-degree angle is \(4.5\) units.
- The side \(y\) adjacent to the 30-degree angle is \(4.5\sqrt{3}\) units.
This analysis uses fundamental trigonometric ratios to determine the lengths of the sides in a right triangle when one angle and the hypotenuse are known.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd98eee17-7906-46be-aec0-2bb2cdc2819f%2Faa58869b-4159-491c-b8dc-72858758f826%2Fnn8tdz_processed.jpeg&w=3840&q=75)
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