Binomial Theorem revision notes

1. Binomial Theorem for Positive Integral Indices

For any positive integer $n$: $(a + b)^n = {}^nC_0 a^n + {}^nC_1 a^{n-1}b + {}^nC_2 a^{n-2}b^2 + \dots + {}^nC_{n-1} a b^{n-1} + {}^nC_n b^n$ Using summation notation, this is expressed as: $\sum_{k=0}^n {}^nC_k a^{n-k} b^k$

Observations on the Expansion:

1. Number of Terms: The total number of terms in the expansion is $(n + 1)$, which is one more than the index.
2. Power Progression: The powers of the first quantity '$a$' go on decreasing by 1 (starting from $n$ to $0$), whereas the powers of the second quantity '$b$' increase by 1 (starting from $0$ to $n$).
3. Sum of Indices: In each term of the expansion, the sum of the indices of $a$ and $b$ is always equal to $n$.

2. Special Cases of the Binomial Expansion

By substituting specific values into the standard expansion, we get the following critical identities:

• Replacing $b$ with $-y$: $(x - y)^n = {}^nC_0 x^n - {}^nC_1 x^{n-1}y + {}^nC_2 x^{n-2}y^2 - \dots + (-1)^n {}^nC_n y^n$
• Setting $a = 1$ and $b = x$: $(1 + x)^n = {}^nC_0 + {}^nC_1 x + {}^nC_2 x^2 + \dots + {}^nC_n x^n$
• Setting $a = 1$ and $b = -x$: $(1 - x)^n = {}^nC_0 - {}^nC_1 x + {}^nC_2 x^2 - \dots + (-1)^n {}^nC_n x^n$
• Sum of all Binomial Coefficients (Setting $x = 1$ in $(1+x)^n$): $2^n = {}^nC_0 + {}^nC_1 + {}^nC_2 + \dots + {}^nC_n$
• Alternating Sum of Binomial Coefficients (Setting $x = 1$ in $(1-x)^n$): $0 = {}^nC_0 - {}^nC_1 + {}^nC_2 - \dots + (-1)^n {}^nC_n$

3. General Term & Specific Terms

4. Properties of Binomial Coefficients

Let $C_r$ denote ${}^nC_r$:

• $C_r = C_{n-r}$ (Symmetry)
• $r \cdot {}^nC_r = n \cdot {}^{n-1}C_{r-1}$ JEE Tip (Crucial for cancelling $r$ in summation problems).
• $\frac{{}^nC_r}{r+1} = \frac{{}^{n+1}C_{r+1}}{n+1}$ (Crucial for integration-based series).
• ${}^nC_r + {}^nC_{r-1} = {}^{n+1}C_r$ (Pascal's Identity, heavily used in proofs).
• $\sum_{r=0}^n C_r^2 = {}^{2n}C_n$
• $\sum_{r=0}^{n-k} C_r \cdot C_{r+k} = {}^{2n}C_{n-k}$

5. Multinomial Theorem

For an expansion of $(x_1 + x_2 + \dots + x_k)^n$:

• General term: $\frac{n!}{p_1! p_2! \dots p_k!} x_1^{p_1} x_2^{p_2} \dots x_k^{p_k}$, subject to $p_1 + p_2 + \dots + p_k = n$.

6. Binomial Theorem for Any Index

If $n$ is a negative integer or a fraction, and $|x| < 1$: $(1+x)^n = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \dots \text{ up to } \infty$

• JEE Tip The series goes to infinity. The concept of $nC_r$ is invalid here because factorials of negative/fractional numbers are undefined in standard elementary combinatorics.

• The combination formula $nC_r = \frac{n!}{r!(n-r)!}$ is ONLY valid for $0 \le r \le n$, where $n$ is a non-negative integer.
• By definition, $nC_0 = 1$ and $nC_n = 1$.
• The standard expansion $(a+b)^n$ strictly assumes $n$ is a positive integer. For fractional/negative indices (JEE Advanced), you must extract the dominant term to force the form $(1+x)^n$ where $|x| < 1$.

• Pascal's Triangle: A structural diagram resembling a triangle. The rows correspond to the index $n$. For example, the row for index 5 is 1 5 10 10 5 1. Using this row, $(2x + 3y)^5$ expands easily by attaching decreasing powers of $(2x)$ and increasing powers of $(3y)$.

Derivation 1: Proof of the Binomial Theorem by Principle of Mathematical Induction

1. Base Case ($n=1$): $(a+b)^1 = {}^1C_0 a^1 + {}^1C_1 b^1 = a+b$. True.
2. Assumption Case ($n=k$): Assume $(a+b)^k = \sum_{r=0}^k {}^kC_r a^{k-r} b^r$.
3. Inductive Step ($n=k+1$): Multiply the assumption by $(a+b)$. $(a+b)^{k+1} = (a+b)({}^kC_0 a^k + {}^kC_1 a^{k-1}b + \dots + {}^kC_k b^k)$ Group like terms: $= {}^kC_0 a^{k+1} + ({}^kC_1 + {}^kC_0)a^k b + ({}^kC_2 + {}^kC_1)a^{k-1}b^2 + \dots + {}^kC_k b^{k+1}$ Using Pascal's Identity $({}^kC_r + {}^kC_{r-1} = {}^{k+1}C_r)$ and ${}^kC_0 = {}^{k+1}C_0 = 1$, the expansion morphs precisely into the definition for index $k+1$.

Problem Solving Approach: Divisibility Problems

Objective: Prove $a^n - b^n$ leaves a specific remainder when divided by a number $P$. Standard Step-by-Step:

1. Express the larger base as the smaller base plus a constant. E.g., to find the remainder of $6^n - 5^n$ mod 25, write $6 = 1+5$.
2. Expand $(1+5)^n$ using Binomial Theorem: ${}^nC_0 + {}^nC_1 5 + {}^nC_2 5^2 + \dots + {}^nC_n 5^n$.
3. Subtract the $5^n$ (or whatever is given in the problem) and observe that all remaining higher power terms contain the divisor (e.g., $5^2 = 25$).
4. Factor out the divisor: $6^n - 5^n = 1 + 25({}^nC_2 + 5 {}^nC_3 + \dots)$.
5. Formulate as $P \cdot k + R$. Remainder is $R=1$.

• JEE Tip If the remainder $R$ turns out to be negative (e.g., $-4$ mod 25), convert it to a positive remainder by adding the divisor: $-4 + 25 = 21$.

Problem Solving Approach: Estimation Problems

Objective: Compare a massive exponent (e.g., $1.01^{1000000}$) to a static number (e.g., $10000$).

1. Split the decimal into $(1 + x)$ where $x$ is small: $(1 + 0.01)^{1000000}$.
2. Expand the first few terms: ${}^nC_0 + {}^nC_1(0.01) + \text{positive terms}$.
3. Evaluate: $1 + 1000000 \times 0.01 = 1 + 10000 = 10001$.
4. Conclude: Since $10001 > 10000$, and all subsequent terms are strictly positive, the exponential expression is strictly larger.

• Term Position vs Index r: The $r$-th term in an expansion is NOT $T_r = {}^nC_r a^{n-r}b^r$. Due to the ${}^nC_0$ term, the $r$-th term is actually $T_r = {}^nC_{r-1} a^{n-(r-1)}b^{r-1}$. Always use $T_{r+1}$.
• Sign Alternate Convention: In $(x-y)^n$, students often try to absorb the minus sign arbitrarily. It is much safer to treat it as $(x + (-y))^n$ and carry the $(-y)^r$ strictly in the general term formula to avoid parity mistakes.
• Bounds of Combinatorics: In series expansions, remember that ${}^nC_r = 0$ for $r > n$ or $r < 0$. Many summation questions rely on you extending the bounds to infinity simply because the terms automatically become zero.
• Rational Index: The formula $(1+x)^n = 1 + nx + \dots$ ONLY converges if $|x| < 1$. If $x = 3$, you cannot use it. You must pull out the dominant term first.

1. Finding the coefficient of a specific power of $x$ (requires equating the exponent of $x$ in the general term to the target power).
2. Summation of Series involving Binomial Coefficients (often using calculus: differentiating $(1+x)^n$ to get $r \cdot {}^nC_r$ series, or integrating for series with $r$ in the denominator).
3. Remainder when a large exponent like $7^{103}$ is divided by 25.
4. Number of rational/irrational terms in expansions like $(2^{1/3} + 3^{1/4})^{100}$.