Binomial Theorem (JEE & CBSE)

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SECTION 1: BASICS AND NOTATION Binomial Expression: An algebraic expression with two terms, e.g., $a+b$, $2x-3y$. Binomial Power: $(a+b)^n$ means $(a+b)$ multiplied by itself $n$ times. Factorial: $n! = n \times (n-1) \times \dots \times 2 \times 1$. $0! = 1$ $n! = n \cdot (n-1)!$ Binomial Coefficient ($nCr$): Definition: $^nC_r = \binom{n}{r} = \frac{n!}{r!(n-r)!}$ Domain: $n$ is a non-negative integer, $0 \le r \le n$. Used for counting combinations. Basic Properties of $nCr$: Symmetry: $^nC_r = ^nC_{n-r}$ (Choosing $r$ items is same as choosing to leave $n-r$ items). Edge Values: $^nC_0 = ^nC_n = 1$. Values: $^nC_1 = n$. Behavior: For a fixed $n$, $^nC_r$ increases with $r$ up to $r = \lfloor n/2 \rfloor$ and then decreases. Sum Formulas involving $nCr$: $\sum_{r=0}^{n} ^nC_r = ^nC_0 + ^nC_1 + \dots + ^nC_n = 2^n$ (Sum of all coefficients in $(1+x)^n$ for $x=1$). $\sum_{r=0}^{n} (-1)^r ^nC_r = ^nC_0 - ^nC_1 + \dots + (-1)^n ^nC_n = 0$ (Sum of coefficients in $(1+x)^n$ for $x=-1$). SECTION 2: PASCAL’S TRIANGLE Construction Rule: Each interior entry is the sum of the two entries directly above it. Edge entries are 1. Relation to $nCr$: The $k$-th row (starting with $k=0$) contains the binomial coefficients $^kC_0, ^kC_1, \dots, ^kC_k$. Relation to $(a+b)^n$ Coefficients: The entries in row $n$ are the coefficients in the expansion of $(a+b)^n$. Rows up to $n=7$: n=0: 1 n=1: 1 1 n=2: 1 2 1 n=3: 1 3 3 1 n=4: 1 4 6 4 1 n=5: 1 5 10 10 5 1 n=6: 1 6 15 20 15 6 1 n=7: 1 7 21 35 35 21 7 1 SECTION 3: BINOMIAL THEOREM FOR POSITIVE INTEGRAL INDEX Statement of Binomial Theorem: $$(a+b)^n = \sum_{r=0}^{n} ^nC_r a^{n-r} b^r$$ Where $n \in \mathbb{N}$ (positive integer). Used to expand expressions of the form $(a+b)^n$. Expanded Form: $$(a+b)^n = ^nC_0 a^n b^0 + ^nC_1 a^{n-1} b^1 + ^nC_2 a^{n-2} b^2 + \dots + ^nC_{n-1} a^1 b^{n-1} + ^nC_n a^0 b^n$$ $$(a+b)^n = a^n + ^nC_1 a^{n-1}b + ^nC_2 a^{n-2}b^2 + \dots + n a b^{n-1} + b^n$$ Conditions: Valid for any real numbers $a, b$ and positive integer $n$. Terms: First term: $^nC_0 a^n = a^n$ Second term: $^nC_1 a^{n-1}b$ General Pattern: Power of $a$ decreases by 1, power of $b$ increases by 1 with each successive term. Sum of powers is always $n$. SECTION 4: GENERAL TERM AND PARTICULAR TERM General Term ($T_{r+1}$): The $(r+1)$-th term in the expansion of $(a+b)^n$ is given by: $$T_{r+1} = ^nC_r a^{n-r} b^r$$ This formula helps find any specific term without writing out the full expansion. How to find terms: Term independent of $x$ (Constant Term): Set the power of $x$ in $T_{r+1}$ to $0$ and solve for $r$. Term containing $x^k$: Set the power of $x$ in $T_{r+1}$ to $k$ and solve for $r$. Coefficient of $x^k$: Find $r$ as above, then the coefficient is what remains after $x^k$ in $T_{r+1}$. Step Template for "find coefficient/term containing $x^k$": Identify $a$, $b$, and $n$ from the given binomial. Write down the general term $T_{r+1} = ^nC_r a^{n-r} b^r$. Substitute $a$ and $b$. Simplify to collect all powers of $x$. Equate the exponent of $x$ to the desired power $k$ (or $0$ for independent term) and solve for $r$. If $r$ is a non-negative integer, substitute $r$ back into $T_{r+1}$ to find the term or its coefficient. SECTION 5: MIDDLE TERM(S) Number of Terms: The expansion of $(a+b)^n$ has $n+1$ terms. Cases for Middle Term(s): If $n$ is even: There is one middle term. Position: $(n/2) + 1$. Term: $T_{(n/2)+1} = ^nC_{n/2} a^{n/2} b^{n/2}$. If $n$ is odd: There are two middle terms. Positions: $(n+1)/2$ and $(n+3)/2$. Terms: $T_{(n+1)/2} = ^nC_{(n-1)/2} a^{(n+1)/2} b^{(n-1)/2}$ and $T_{(n+3)/2} = ^nC_{(n+1)/2} a^{(n-1)/2} b^{(n+1)/2}$. Typical Middle-Term Question Patterns: Find the middle term directly, find the coefficient of the middle term, or questions involving properties of the middle term(s). SECTION 6: SPECIAL VALUES AND SUMS OF COEFFICIENTS Sum of all coefficients of $(a+b)^n$: Substitute $a=1, b=1$ into the expansion. $$(1+1)^n = 2^n$$ This gives $\sum_{r=0}^{n} ^nC_r$. Sum of coefficients of even powers and odd powers of $x$ (in $f(x)$ expansion): Let $f(x)$ be the expansion, e.g., $(1+x)^n$. Sum of coefficients of even powers of $x$: $\frac{f(1) + f(-1)}{2}$. Sum of coefficients of odd powers of $x$: $\frac{f(1) - f(-1)}{2}$. Coefficients of $(1+x)^n$ vs $(1-x)^n$: $(1+x)^n = ^nC_0 + ^nC_1 x + ^nC_2 x^2 + \dots + ^nC_n x^n$. $(1-x)^n = ^nC_0 - ^nC_1 x + ^nC_2 x^2 - \dots + (-1)^n ^nC_n x^n$. (Alternating signs for odd powers of $x$). SECTION 7: IMPORTANT IDENTITIES INVOLVING $nCr$ Symmetry Identity: $^nC_r = ^nC_{n-r}$. (Choosing $r$ items is equivalent to not choosing $n-r$ items). Pascal's Identity (Adjacent Sum): $^nC_r + ^nC_{r-1} = ^{n+1}C_r$. (Used in Pascal's Triangle construction). Multiplicative Form: $^nC_r = \frac{n}{r} \cdot ^{n-1}C_{r-1}$. (Useful for simplifying expressions with $r \cdot ^nC_r$). Some Common Sums: $\sum_{r=0}^{n} ^nC_r = 2^n$. $\sum_{r=0}^{n} r \cdot ^nC_r = n \cdot 2^{n-1}$. (Obtained by differentiating $(1+x)^n$ and setting $x=1$). $\sum_{r=0}^{n} r^2 \cdot ^nC_r = n(n+1)2^{n-2}$ (for $n \ge 1$). Can be derived from $r^2 = r(r-1) + r$. $\sum_{r=0}^{\lfloor n/2 \rfloor} ^nC_{2r} = ^nC_0 + ^nC_2 + ^nC_4 + \dots = 2^{n-1}$. (Sum of even coefficients). $\sum_{r=0}^{\lfloor (n-1)/2 \rfloor} ^nC_{2r+1} = ^nC_1 + ^nC_3 + ^nC_5 + \dots = 2^{n-1}$. (Sum of odd coefficients). $\sum_{r=0}^{n} (-1)^r ^nC_r = 0$. Many identities can be derived by differentiating or integrating binomial expansions like $(1+x)^n$ or $(1-x)^n$ and substituting specific values for $x$. SECTION 8: RATIO OF TERMS AND NUMERICALLY GREATEST TERM (NGT) Ratio of Consecutive Terms: For $(a+b)^n$, the ratio of the $(r+1)$-th term to the $r$-th term is: $$\frac{T_{r+1}}{T_r} = \frac{^nC_r a^{n-r} b^r}{^nC_{r-1} a^{n-r+1} b^{r-1}} = \frac{n-r+1}{r} \cdot \frac{b}{a}$$ This is used to compare adjacent terms. Condition to Locate Numerically Greatest Term (NGT) of $(1+x)^n$: To find the NGT, we find $r$ such that $|T_{r+1}| \ge |T_r|$, i.e., $\left| \frac{T_{r+1}}{T_r} \right| \ge 1$. $$\left| \frac{n-r+1}{r} x \right| \ge 1$$ Solving this inequality for $r$ gives $r \le \frac{(n+1)|x|}{1+|x|}$. Let $m = \frac{(n+1)|x|}{1+|x|}$. If $m$ is an integer, then $T_m$ and $T_{m+1}$ are numerically equal and are the greatest terms. If $m$ is not an integer, then $T_{\lfloor m \rfloor + 1}$ is the unique numerically greatest term. Example Outline (NGT): Find the numerically greatest term in the expansion of $(1 + \frac{3}{4})^{10}$. Here $n=10$, $x=3/4$. Calculate $m = \frac{(10+1)(3/4)}{1+3/4} = \frac{11 \times 3/4}{7/4} = \frac{33}{7} \approx 4.71$. Since $m$ is not an integer, the NGT is $T_{\lfloor 4.71 \rfloor + 1} = T_{4+1} = T_5$. SECTION 9: APPROXIMATIONS USING BINOMIAL THEOREM Method: Express numbers close to a power of 10 (or other convenient base) as a binomial $(X \pm \delta)^n$. Examples: $99^5 = (100-1)^5$, $101^4 = (100+1)^4$, $1001^3 = (1000+1)^3$. Approximation: Use the first few terms of the expansion. For small $\delta/X$, higher powers of $(\delta/X)$ become very small and can be ignored for approximation. Example: $(1.01)^{10} = (1+0.01)^{10} \approx 1 + 10(0.01) + \frac{10 \times 9}{2}(0.01)^2 = 1 + 0.1 + 0.0045 = 1.1045$. Typical Exam Patterns: Questions involving $(1+x)^n$ or $(1-x)^n$ where $x$ is small, to find approximate values or compare magnitudes. SECTION 10: GENERAL TERM IN SPECIAL FORMS General term when base is not $(a+b)$ directly: For $(2x-3)^n$: Treat $a=2x$ and $b=-3$. $T_{r+1} = ^nC_r (2x)^{n-r} (-3)^r$. For $(ax + b/x)^n$: Treat $a=ax$ and $b=b/x$. $T_{r+1} = ^nC_r (ax)^{n-r} (b/x)^r$. Finding Term Independent of $x$ or Term containing $x^k$ in $(ax + b/x)^n$: Write $T_{r+1} = ^nC_r (ax)^{n-r} (b/x)^r = ^nC_r a^{n-r} x^{n-r} b^r x^{-r}$. Collect powers of $x$: $T_{r+1} = ^nC_r a^{n-r} b^r x^{n-2r}$. For term independent of $x$, set $n-2r = 0$ and solve for $r$. For term containing $x^k$, set $n-2r = k$ and solve for $r$. Substitute $r$ back to find the term or its coefficient. SECTION 11: BINOMIAL COEFFICIENT PROPERTIES (COMBINATORIAL VIEW) Link to Counting: $^nC_r$ represents the number of ways to choose $r$ distinct objects from a set of $n$ distinct objects. Simple Identities Explained Combinatorially: $^nC_r = ^nC_{n-r}$: Choosing $r$ objects is the same as choosing which $n-r$ objects to leave behind. $^nC_r + ^nC_{r-1} = ^{n+1}C_r$: Consider choosing $r$ people from $n+1$ people (one of whom is a specific person, say, 'A'). Either 'A' is chosen ($^{n}C_{r-1}$ ways) or 'A' is not chosen ($^{n}C_r$ ways). SECTION 12: NEGATIVE / FRACTIONAL INDEX (JEE NOTE) General Binomial Series: For any real number $n$ (not necessarily a positive integer) and $|x| Key Points: For Class 11 NCERT, focus is on positive integral indices where the expansion is finite. This infinite series formula is crucial for JEE-level approximations, especially involving roots or negative powers (e.g., $\sqrt{1+x}$, $(1-x)^{-1}$). SECTION 13: TYPICAL QUESTION TYPES Expand a given binomial expression using the Binomial Theorem. Find a particular term (e.g., 5th term), middle term(s). Find the term independent of $x$ or the coefficient of $x^k$. Compare magnitudes of expressions like $(1.01)^{100}$ vs $1.1$. Simplify sums involving binomial coefficients using identities. Find the numerically greatest term in an expansion. Simplify expressions of the form $(a+b)^n + (a-b)^n$ or $(a-b)^n - (a+b)^n$. Problems involving properties of coefficients (e.g., sum of coefficients, ratio of coefficients).