🔢 Topic Summary
Counting techniques are essential tools in probability for determining the number of possible outcomes in various situations. They help us calculate probabilities by providing a systematic way to count favorable outcomes and the total number of outcomes. These techniques include permutations, combinations, the fundamental counting principle, and more. Understanding these techniques allows us to solve a wide range of probability problems, from simple coin flips to complex arrangements and selections.
The fundamental counting principle states that if there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both. Permutations count arrangements where order matters, while combinations count selections where order doesn't matter. Each technique has specific formulas and conditions for their application. Practice is key to mastering these concepts and applying them effectively!
🧮 Part A: Vocabulary
Match the terms with their correct definitions:
| Term |
Definition |
| 1. Permutation |
A. A selection of items where order does not matter. |
| 2. Combination |
B. The set of all possible outcomes of an experiment. |
| 3. Sample Space |
C. An arrangement of items in a specific order. |
| 4. Factorial |
D. The number of ways to choose $r$ items from a set of $n$ items without regard to order. |
| 5. nCr |
E. The product of all positive integers less than or equal to a given positive integer. |
➕ Part B: Fill in the Blanks
The __________ counting principle states that if there are $m$ ways to do one thing and $n$ ways to do another, then there are $m \times n$ ways to do both. A __________ is an arrangement of objects in a specific order. In contrast, a __________ is a selection of objects where order does not matter. The formula for permutations is $P(n, r) = \frac{n!}{(n-r)!}$, and the formula for combinations is $C(n, r) = \frac{n!}{r!(n-r)!}$. The symbol $n!$ represents the __________ of $n$. When calculating probabilities, it's essential to consider whether __________ is important to choose the correct counting technique.
🤔 Part C: Critical Thinking
Explain, in your own words, how the fundamental counting principle, permutations, and combinations can be used together to solve a complex probability problem. Provide a specific example to illustrate your explanation.