Summary and Quick Revision

The Chapter in One Idea

Everything here flows from one line:

$(a+b)^n = \sum_{k=0}^{n} {^nC_k}\,a^{n-k}b^k$ — combinations count the coefficients. Each term picks $k$ factors to contribute $b$; there are $^nC_k$ ways to pick them. Pascal's triangle, the special cases, the numerical tricks and the general term are all this sum read in different ways.

The Theorem and Its Structure — Formula Card

• Statement: $(a+b)^n = {^nC_0}a^n + {^nC_1}a^{n-1}b + \cdots + {^nC_n}b^n$; proved by induction using $^kC_r + {^kC_{r-1}} = {^{k+1}C_r}$.
• Bookkeeping: $n + 1$ terms; powers of $a$ fall $n \to 0$, powers of $b$ rise $0 \to n$; every term's indices sum to $n$.
• Pascal's triangle (Meru Prastara): row $n$ is $^nC_0, \ldots, {^nC_n}$; each entry is the sum of the two above. Memorise rows to index 6.
• Special cases: $(x-y)^n$ alternates signs starting positive; $(1+x)^n = \sum {^nC_r}x^r$; $(1-x)^n$ alternates.
• Identities (put $x = \pm 1$): $\sum {^nC_r} = 2^n$; alternating sum $= 0$; even- and odd-position coefficient sums are each $2^{n-1}$.
• The bases carry their coefficients: $(2x)^3 = 8x^3$ — the whole first quantity is raised to the power.

Applications — Formula Card

• Numerical powers: write the base as $100 \pm k$ (or $1 \pm x$): $(98)^5 = (100-2)^5 = 9039207968$; $(101)^4 = 104060401$ shows the row $1,4,6,4,1$ in its digits.
• Approximations: keep the first few terms — $(0.99)^5 \approx 1 - 0.05 + 0.001 = 0.951$.
• Comparisons: for $x > 0$, $(1+x)^n > 1 + nx$ — settles $(1.01)^{1000000} > 10000$ and $(1.1)^{10000} > 1000$ in two terms.
• Divisibility/remainders: split so the modulus appears squared from the third term: $6^n = (1+5)^n$ gives $6^n - 5n \equiv 1 \pmod{25}$; $9^{n+1} - 8n - 9$ is divisible by 64; $8^n - 7n \equiv 1 \pmod{49}$.
• Conjugate pairs: $(a+b)^n + (a-b)^n$ keeps even powers of $b$ (doubled); the difference keeps odd powers. With surd $b$, sums become integers: $(\sqrt{2}+1)^6 + (\sqrt{2}-1)^6 = 198$.
• Factor results: $a^n = (a-b+b)^n$ shows $(a-b)$ divides $a^n - b^n$.

JEE Quick Card — General and Middle Terms

• General term: $T_{r+1} = {^nC_r}\,a^{n-r}b^r$ — the term number is one more than $r$.
• Coefficient of $x^k$: write $T_{r+1}$, collect the power of $x$ in terms of $r$, set equal to $k$, solve, substitute. Non-integer or out-of-range $r$ means the term does not exist (coefficient 0).
• Term independent of $x$: set the collected power to zero.
• Middle terms: $n$ even → single middle $T_{\frac{n}{2}+1}$; $n$ odd → two middles $T_{\frac{n+1}{2}}, T_{\frac{n+3}{2}}$. Greatest binomial coefficient of $(1+x)^{2n}$: $^{2n}C_n$.
• Sum of coefficients: substitute $x = 1$ (all variables to 1). Distinguish from the sum of binomial coefficients, which is always $2^n$.
• Term from the end: $r$-th from the end $=$ $(n - r + 2)$-th from the start.
• Equal coefficients: $^nC_a = {^nC_b}$ gives $a = b$ or $a + b = n$ — check both, then reject out-of-range roots.

Last-Minute Mistake Checklist

Before the exam, scan this list — each item is a real mark lost by real students:

1. $(2x)^4 = 16x^4$, not $2x^4$ — the whole base carries the power.
2. Term count: index $n$ means $n + 1$ terms; miscounting loses or doubles a term.
3. Sign slips in $(x - y)^n$: signs alternate starting with $+$; the term with $y^r$ carries $(-1)^r$.
4. $T_{r+1}$ vs $T_r$: the 7th term has $r = 6$. Off-by-one here wrecks the whole answer.
5. Solving for $r$ and accepting a non-integer — that means NO such term (coefficient 0), not a rounding.
6. Middle terms: even $n$ has ONE middle; odd $n$ has TWO. Don't quote $T_{n/2}$ for odd $n$.
7. "Sum of coefficients" (substitute $x = 1$ in everything) confused with "sum of binomial coefficients" ($2^n$).
8. In conjugate pairs, the SUM keeps EVEN powers of $b$; the DIFFERENCE keeps ODD. Swapping them flips the answer.
9. Divisibility proofs: split off the right base so the modulus appears squared — mod 25 use 5's, mod 64 use 8's, mod 49 use 7's.
10. In numerical work with a minus split, signs alternate — writing all plus signs for $(100 - 2)^5$ is the classic slip.