Binomial Theorem: Expansion, Pascal’s Triangle, Applications

TAKEAWAYS:

• Understand the Binomial Theorem formula and how to expand expressions like $(a+b{)}^n$.
• Use Pascal’s Triangle to find coefficients quickly for small exponents.
• Apply the theorem to business problems like compound interest, profit calculations, and risk analysis.
• Learn special cases (e.g., $(a-b{)}^n$, $(1+x{)}^n$) and their shortcuts.
• Master term selection to find specific terms in expansions without writing the full expansion.
• Practice NEB-style questions to score full marks in exams.

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What is the Binomial Theorem?

The Binomial Theorem helps expand expressions of the form $(a+b{)}^n$ where $n$ is a positive integer. It is widely used in business for calculations involving growth, probability, and financial modeling.

Key Terms:

• Binomial: An expression with two terms, e.g., $(x+y)$.
• Exponent ($n$): The power to which the binomial is raised.
• Term: Each part of the expansion, e.g., $a{x}^3{y}^2$ in $(x+y{)}^5$.

Formula:

The expansion of $(a+b{)}^n$ is given by: $$(a+b{)}^n=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})a^{n-k}{b}^k$$ where $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ (read as "n choose k") is the binomial coefficient, calculated as: $$\left(\genfrac{}{}{0ex}{}{n}{k}\right)=\frac{n!}{k!(n-k)!}$$

Example 1: Expand $(x+2{)}^4$

Let’s expand this step-by-step using the Binomial Theorem.

1. Identify $a=x$, $b=2$, and $n=4$.
2. Write the expansion using the formula: $$(x+2{)}^4=\sum _{k=0}^4(\genfrac{}{}{0ex}{}{4}{k})x^{4-k}{2}^k$$
3. Calculate each term:
   • For $k=0$: $\left(\genfrac{}{}{0ex}{}{4}{0}\right)x^4{2}^0=1\cdot x^4\cdot 1=x^4$
   • For $k=1$: $\left(\genfrac{}{}{0ex}{}{4}{1}\right)x^3{2}^1=4\cdot x^3\cdot 2=8x^3$
   • For $k=2$: $\left(\genfrac{}{}{0ex}{}{4}{2}\right)x^2{2}^2=6\cdot x^2\cdot 4=24x^2$
   • For $k=3$: $\left(\genfrac{}{}{0ex}{}{4}{3}\right)x^1{2}^3=4\cdot x\cdot 8=32x$
   • For $k=4$: $\left(\genfrac{}{}{0ex}{}{4}{4}\right)x^0{2}^4=1\cdot 1\cdot 16=16$
4. Combine all terms: $$(x+2{)}^4=x^4+8x^3+24x^2+32x+16$$

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Pascal’s Triangle and Binomial Coefficients

Pascal’s Triangle is a simple way to find binomial coefficients without calculating factorials. Each number in the triangle corresponds to a binomial coefficient $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$.

How to Build Pascal’s Triangle:

1. Start with 1 at the top.
2. Each subsequent row starts and ends with 1.
3. Each interior number is the sum of the two numbers directly above it.

Example 2: Find the 5th row of Pascal’s Triangle

The 5th row (starting from row 0) is: $$1,5,10,10,5,1$$ This means the coefficients for $(a+b{)}^5$ are 1, 5, 10, 10, 5, 1.

Using Pascal’s Triangle to Expand $(3x-y{)}^4$:

1. Write the coefficients from the 4th row of Pascal’s Triangle: 1, 4, 6, 4, 1.
2. Apply the signs alternately (since the second term is $-y$): $$(3x-y{)}^4=1\cdot (3x{)}^4(-y{)}^0+4\cdot (3x{)}^3(-y{)}^1+6\cdot (3x{)}^2(-y{)}^2+4\cdot (3x{)}^1(-y{)}^3+1\cdot (3x{)}^0(-y{)}^4$$
3. Simplify each term:
   • $1\cdot 81x^4\cdot 1=81x^4$
   • $4\cdot 27x^3\cdot (-y)=-108x^{3}y$
   • $6\cdot 9x^2\cdot y^2=54x^2{y}^2$
   • $4\cdot 3x\cdot (-y{)}^3=-12x{y}^3$
   • $1\cdot 1\cdot y^4=y^4$
4. Combine all terms: $$(3x-y{)}^4=81x^4-108x^{3}y+54x^2{y}^2-12x{y}^3+y^4$$

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Special Cases of the Binomial Theorem

The Binomial Theorem has shortcuts for specific forms:

Example 3: Expand $(2x^2-\frac{1}{x}{)}^3$ using $(a-b{)}^n$

1. Identify $a=2x^2$, $b=\frac{1}{x}$, and $n=3$.
2. Use the formula for $(a-b{)}^3$: $$(a-b{)}^3=a^3-3a^{2}b+3a{b}^2-b^3$$
3. Substitute $a$ and $b$: $$(2x^2{)}^3-3(2x^2{)}^2\left(\frac{1}{x}\right)+3(2x^2){\left(\frac{1}{x}\right)}^2-{\left(\frac{1}{x}\right)}^3$$
4. Simplify each term:
   • $8x^6$
   • $-3\cdot 4x^4\cdot \frac{1}{x}=-12x^3$
   • $3\cdot 2x^2\cdot \frac{1}{x^2}=6$
   • $-\frac{1}{x^3}$
5. Combine all terms: $$(2x^2-\frac{1}{x}{)}^3=8x^6-12x^3+6-\frac{1}{x^3}$$

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Finding Specific Terms in Binomial Expansion

Instead of expanding the entire expression, you can find a specific term using the general term formula: $$T_{k+1}=\left(\genfrac{}{}{0ex}{}{n}{k}\right)a^{n-k}{b}^k$$ where $T_{k+1}$ is the $(k+1)$th term.

Example 4: Find the 4th term in $(3x^2+\frac{1}{x}{)}^5$

1. Identify $a=3x^2$, $b=\frac{1}{x}$, $n=5$, and $k=3$ (since the 4th term corresponds to $k=3$).
2. Use the general term formula: $$T_4=\left(\genfrac{}{}{0ex}{}{5}{3}\right)(3x^2{)}^{5-3}{\left(\frac{1}{x}\right)}^3$$
3. Calculate the binomial coefficient and simplify: $$\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$$ $$(3x^2{)}^2=9x^4$$ $${\left(\frac{1}{x}\right)}^3=\frac{1}{x^3}$$
4. Combine all parts: $$T_4=10\cdot 9x^4\cdot \frac{1}{x^3}=90x$$

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Applications of the Binomial Theorem in Business

The Binomial Theorem is useful in:

1. Finance: Calculating compound interest, annuities, and investment growth.
2. Probability: Modeling risk and uncertainty in business decisions.
3. Marketing: Analyzing consumer behavior and sales trends.
4. Operations: Optimizing production and resource allocation.

Example 5: Compound Interest Calculation

Suppose you invest $1000 at an annual interest rate of 5% compounded annually. What is the amount after 3 years?

1. The formula for compound interest is: $$A=P(1+r{)}^n$$ where $P=1000$, $r=0.05$, and $n=3$.
2. Expand $(1.05{)}^3$ using the Binomial Theorem: $$(1.05{)}^3=1+3(1.05)+3(1.05{)}^2+(1.05{)}^3$$ $$=1+3.15+3.255+1.157625\approx 8.562625$$
3. Calculate the final amount: $$A=1000\times 8.562625=\$8562.63$$

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Common Mistakes to Avoid

1. Incorrect Binomial Coefficients: Forgetting to use $\left(\genfrac{}{}{0ex}{}{n}{k}\right)$ or miscalculating factorials.
2. Sign Errors: Forgetting to alternate signs in expansions like $(a-b{)}^n$.
3. Exponent Rules: Misapplying exponents when simplifying terms (e.g., $(x^2{)}^3=x^6$, not $x^5$).
4. Term Selection: Confusing the term number with the exponent $k$ (remember $T_{k+1}$ corresponds to $k$).

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NEB Board-Style Questions

Short Answer Questions

1. Expand $(x+\frac{1}{x}{)}^4$ using the Binomial Theorem.
2. Find the 5th term in the expansion of $(2x-\frac{1}{x^2}{)}^6$.
3. Write the first three terms in the expansion of $(1-3x{)}^5$.
4. Use Pascal’s Triangle to expand $(a+b{)}^4$.

Long Answer Questions

1. Expand $(3x^2-\frac{2}{x}{)}^4$ and find the term independent of $x$.

   • Solution:
     • Use the Binomial Theorem to expand.
     • Identify the term where the powers of $x$ cancel out (i.e., $x^0$).
     • The term independent of $x$ is $6\cdot (3x^2{)}^2\cdot {\left(\frac{2}{x}\right)}^2=6\cdot 9x^4\cdot \frac{4}{x^2}=216x^2$. Correction: Actually, the term independent of $x$ occurs when the exponents of $x$ in the term cancel out. For $(3x^2-\frac{2}{x}{)}^4$, the general term is: $$T_{k+1}=\left(\genfrac{}{}{0ex}{}{4}{k}\right)(3x^2{)}^{4-k}{\left(-\frac{2}{x}\right)}^k=(\genfrac{}{}{0ex}{}{4}{k})3^{4-k}(-2{)}^k{x}^{8-3k}$$ For the term independent of $x$, set $8-3k=0$: $$8-3k=0⟹k=\frac{8}{3}$$ Since $k$ must be an integer, there is no term independent of $x$ in this expansion. Note: This is a trick question! Always verify.

2. A business invests $5000 at an annual interest rate of 4%. Calculate the amount after 3 years using the Binomial Theorem for compound interest.

   • Solution:
     • Use $A=P(1+r{)}^n$ where $P=5000$, $r=0.04$, $n=3$.
     • Expand $(1.04{)}^3$: $$(1.04{)}^3=1+3(1.04)+3(1.04{)}^2+(1.04{)}^3$$ $$=1+3.12+3.2496+1.07488\approx 8.44448$$
     • Calculate $A$: $$A=5000\times 8.44448\approx \$42,222.40$$

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Exam Tip

1. Memorize the Formula: Know the Binomial Theorem formula and how to apply it.
2. Practice Pascal’s Triangle: Be comfortable using it for small exponents (up to $n=6$).
3. Term Selection: Learn to find specific terms without full expansion to save time.
4. Check for Errors: Always verify signs, exponents, and coefficients.
5. Business Applications: Relate the theorem to real-world problems like interest calculations or probability.
6. NEB Patterns: Expect questions on expansion, term selection, and applications in finance or probability.

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Summary Table: Binomial Theorem Key Points

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Final Notes

• The Binomial Theorem is a powerful tool for expanding expressions and solving real-world problems.
• Practice regularly to master term selection and expansion techniques.
• Always double-check your calculations, especially signs and exponents.
• Relate the theorem to business scenarios to understand its practical importance.

Good luck with your NEB exams! Keep practicing, and you’ll master this topic. 🚀