How to find a specific term in a binomial expansion using nCr

📚 Understanding Binomial Expansion and the General Term

Binomial expansion is a method of expanding expressions of the form $(a + b)^n$, where $n$ is a positive integer. The general term in the expansion helps us find a specific term without having to expand the entire expression. This is where the binomial coefficient, often expressed as 'nCr' or $\binom{n}{r}$, comes in handy.

📜 Historical Context

The binomial theorem, which governs binomial expansions, has roots stretching back to ancient mathematics. Mathematicians like Isaac Newton significantly contributed to its generalization. Pascal's Triangle, while predating Newton, offers a visual and numerical way to understand binomial coefficients, linking combinatorics to algebra.

🔑 Key Principles and the nCr Formula

The binomial theorem states that for any non-negative integer $n$:

$\qquad (a + b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^r$

Here, $\binom{n}{r}$ represents the binomial coefficient, also written as $nCr$, and is calculated as:

$\qquad \binom{n}{r} = \frac{n!}{r!(n-r)!}$

• 🧮 n: Represents the power to which the binomial is raised.
• 🔢 r: Represents the term number (starting from 0). Note that $r$ starts from 0, so to find the $k^{th}$ term, you would use $r = k - 1$.
• ➗ n!: (n factorial) is the product of all positive integers up to $n$. For example, $5! = 5 \times 4 \times 3 \times 2 \times 1 = 120$.
• 💡 Key Insight: The $(r+1)^{th}$ term in the expansion of $(a+b)^n$ is given by $T_{r+1} = \binom{n}{r} a^{n-r} b^r$.

✍️ Step-by-Step Guide to Finding a Specific Term

1. Identify n, a, and b: Determine the values of $n$, $a$, and $b$ from the given binomial expression $(a + b)^n$.
2. Determine r: If you want to find the $k^{th}$ term, set $r = k - 1$.
3. Calculate nCr: Use the formula $\binom{n}{r} = \frac{n!}{r!(n-r)!}$ to find the binomial coefficient.
4. Apply the formula: Substitute $n$, $r$, $a$, and $b$ into the general term formula $T_{r+1} = \binom{n}{r} a^{n-r} b^r$ and simplify.

➗ Example 1: Finding the 4th term of $(x + 2)^6$

Here, $a = x$, $b = 2$, and $n = 6$. We want to find the 4th term, so $r = 4 - 1 = 3$.

$\qquad \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6!}{3!3!} = \frac{720}{6 \times 6} = 20$

Now, apply the formula:

$\qquad T_4 = \binom{6}{3} x^{6-3} (2)^3 = 20 \cdot x^3 \cdot 8 = 160x^3$

Therefore, the 4th term is $160x^3$.

➕ Example 2: Finding the middle term of $(3x - 1)^{8}$

Here, $a = 3x$, $b = -1$, and $n = 8$. Since $n = 8$ is even, there is one middle term, which is the $(\frac{8}{2} + 1)^{th} = 5^{th}$ term. So, $r = 5 - 1 = 4$.

$\qquad \binom{8}{4} = \frac{8!}{4!(8-4)!} = \frac{8!}{4!4!} = \frac{40320}{24 \times 24} = 70$

Now, apply the formula:

$\qquad T_5 = \binom{8}{4} (3x)^{8-4} (-1)^4 = 70 \cdot (3x)^4 \cdot 1 = 70 \cdot 81x^4 = 5670x^4$

Therefore, the middle term is $5670x^4$.

➖ Example 3: Finding the term independent of x in $(2x + \frac{1}{x})^6$

Here, $a = 2x$, $b = \frac{1}{x}$, and $n = 6$. We need to find $r$ such that the power of $x$ is 0.

The general term is: $T_{r+1} = \binom{6}{r} (2x)^{6-r} (\frac{1}{x})^r = \binom{6}{r} 2^{6-r} x^{6-r} x^{-r} = \binom{6}{r} 2^{6-r} x^{6-2r}$

For the term to be independent of $x$, we need $6 - 2r = 0$, which gives $r = 3$.

$\qquad \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{720}{6 \times 6} = 20$

Now, apply the formula:

$\qquad T_4 = \binom{6}{3} (2x)^{6-3} (\frac{1}{x})^3 = 20 \cdot (2x)^3 \cdot (\frac{1}{x})^3 = 20 \cdot 8x^3 \cdot \frac{1}{x^3} = 160$

Therefore, the term independent of $x$ is 160.

📝 Practice Quiz

Find the specified term in each binomial expansion:

1. ❓ Find the 3rd term of $(x + 3)^5$
2. ➕ Find the 6th term of $(2x - 1)^7$
3. ➗ Find the middle term of $(x^2 + 2)^4$
4. 💡 Find the term independent of x in $(x - \frac{2}{x})^8$

✅ Solutions

1. 3rd term of $(x + 3)^5$: $90x^3$
2. 6th term of $(2x - 1)^7$: $-672x^2$
3. Middle term of $(x^2 + 2)^4$: $24x^4$
4. Term independent of x in $(x - \frac{2}{x})^8$: $1120$

🌍 Real-World Applications

• 📈 Probability: Binomial expansions are used to calculate probabilities in scenarios with binary outcomes (success/failure).
• 💰 Finance: They appear in financial models, such as option pricing.
• 💻 Computer Science: Used in algorithms and data analysis.

🎓 Conclusion

Understanding how to find a specific term in a binomial expansion using nCr is crucial for various mathematical and real-world applications. By mastering the formula and practicing with examples, you can confidently tackle these problems. Keep practicing, and you'll become a binomial expansion pro in no time!