Solved examples of Binomial Theorem expansion using nCr step-by-step

📚 Quick Study Guide

• 🔢 The Binomial Theorem allows us to expand expressions of the form $(a + b)^n$.
• ➕ The general term in the expansion of $(a + b)^n$ is given by: $T_{r+1} = {}^nC_r \cdot a^{n-r} \cdot b^r$, where $r = 0, 1, 2, ..., n$.
• 🧮 The binomial coefficient ${}^nC_r$ (also written as $\binom{n}{r}$) is calculated as: ${}^nC_r = \frac{n!}{r!(n-r)!}$, where $n!$ represents the factorial of $n$.
• 💡Remember that $0! = 1$.
• 📝 The values of ${}^nC_r$ are symmetrical, meaning ${}^nC_r = {}^nC_{n-r}$. This can save you computation time!

Practice Quiz

1. What is the value of ${}^5C_2$?

   1. 5
   2. 10
   3. 12
   4. 20

2. What is the coefficient of $x^3$ in the expansion of $(x + 2)^5$?

   1. 10
   2. 40
   3. 80
   4. 160

3. What is the constant term in the expansion of $(2x + \frac{1}{x})^6$?

   1. 15
   2. 20
   3. 60
   4. 160

4. Find the middle term in the expansion of $(x - 1)^{10}$.

   1. ${}^{10}C_5 x^5$
   2. -${}^{10}C_5 x^5$
   3. ${}^{10}C_5 x^5 (-1)^5$
   4. -${}^{10}C_5 $

5. What is the value of ${}^nC_0 + {}^nC_n$?

   1. 0
   2. 1
   3. 2
   4. n

6. In the expansion of $(a + b)^n$, if the coefficients of the second and third terms are 20 and 190 respectively, what is the value of n?

   1. 10
   2. 20
   3. 19
   4. No such 'n' exists.

7. Determine the sum of the coefficients in the binomial expansion of $(2x - 1)^{5}$.

   1. 0
   2. 1
   3. 32
   4. -1