Pre-Calculus test questions on Binomial Coefficients and expansion

📚 Quick Study Guide

🔢 Binomial Coefficient: Denoted as $\binom{n}{k}$ (read as "n choose k"), it represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to order. The formula is: $\binom{n}{k} = \frac{n!}{k!(n-k)!}$, where $n!$ is the factorial of $n$. 🧮 Factorial: The factorial of a non-negative integer $n$, denoted by $n!$, is the product of all positive integers less than or equal to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$. 💡 Binomial Theorem: This theorem describes the algebraic expansion of powers of a binomial. For any positive integer $n$, the theorem states: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k = \binom{n}{0}a^n + \binom{n}{1}a^{n-1}b + \binom{n}{2}a^{n-2}b^2 + ... + \binom{n}{n}b^n$. 📝 Pascal's Triangle: A triangular array where each number is the sum of the two numbers above it. The rows of Pascal's triangle represent the coefficients in the binomial expansion. The $k$-th entry in the $n$-th row is $\binom{n}{k}$. ➕ Properties of Binomial Coefficients:
✅ Symmetry: $\binom{n}{k} = \binom{n}{n-k}$ ✅ Sum of Coefficients: $\sum_{k=0}^{n} \binom{n}{k} = 2^n$

🧪 Practice Quiz

1. What is the value of $\binom{8}{3}$?
   1. 36
   2. 56
   3. 24
   4. 84
2. What is the coefficient of $x^5$ in the expansion of $(x+2)^8$?
   1. 56
   2. 112
   3. 1792
   4. 448
3. Which of the following is equal to $\binom{n}{2}$?
   1. $\frac{n(n+1)}{2}$
   2. $\frac{n(n-1)}{2}$
   3. $\frac{n!}{2!}$
   4. $\frac{(n-2)!}{n!}$
4. What is the sum of the binomial coefficients in the expansion of $(a+b)^7$?
   1. 7
   2. 64
   3. 128
   4. 49
5. What is the middle term in the expansion of $(x+y)^6$?
   1. $20x^3y^3$
   2. $6x^3y^3$
   3. $15x^3y^3$
   4. $\binom{6}{3}$
6. Simplify: $\binom{n}{1} + \binom{n}{n-1}$?
   1. 1
   2. 2
   3. $2n$
   4. $n^2$
7. What is the constant term in the expansion of $(x + \frac{1}{x})^4$?
   1. 1
   2. 4
   3. 6
   4. 8