Practical Examples of Pascal's Triangle in Probability & Combinatorics

📚 Quick Study Guide

• 📐 Pascal's Triangle: A triangular array where each number is the sum of the two numbers above it. The edges are always 1.
• 🔢 Combinations: Pascal's Triangle provides the values for combinations, denoted as $nCr$ or $\binom{n}{r}$, which represents the number of ways to choose $r$ items from a set of $n$ items without regard to order.
• ➕ Formula for Combinations: The combination formula is $nCr = \frac{n!}{r!(n-r)!}$, where $n!$ is the factorial of $n$.
• 🎲 Probability: Pascal's Triangle helps determine probabilities in scenarios with equally likely outcomes, such as coin flips or dice rolls.
• 🔗 Relationship: The $r^{th}$ entry in the $n^{th}$ row of Pascal's Triangle (starting with row 0 and entry 0) gives the value of $nCr$.
• 📈 Binomial Theorem: Pascal's Triangle provides the coefficients in the binomial expansion of $(a + b)^n$.

🧪 Practice Quiz

1. What is the value of the 4th element (index 3) in the 6th row (index 5) of Pascal's Triangle?
1. 10
2. 15
3. 20
4. 25
2. A coin is flipped 5 times. What is the probability of getting exactly 3 heads? (Hint: Use Pascal's Triangle to find the number of ways to get 3 heads)
1. 5/16
2. 1/2
3. 3/8
4. 1/4
3. In how many ways can a committee of 3 people be chosen from a group of 7 people?
1. 21
2. 35
3. 42
4. 120
4. What is the coefficient of $x^2$ in the expansion of $(x+1)^4$?
1. 1
2. 4
3. 6
4. 10
5. You have a bag with 6 different colored balls. You want to pick 2 balls. How many different combinations of 2 balls can you pick?
1. 6
2. 12
3. 15
4. 30
6. What is the sum of all the numbers in the 5th row (index 4) of Pascal's Triangle?
1. 8
2. 16
3. 24
4. 32
7. What is the value of $^8C_2$ (8 choose 2)?
1. 16
2. 28
3. 56
4. 112