Chapter 7: Binomial Theorem

How would you expand $(1 + x)^{20}$ without multiplying twenty copies by hand? The binomial theorem answers this in a single line, giving every term and every coefficient by formula.

For positive integer exponents the formula is $(a + b)^n = \sum_{r = 0}^{n} \binom{n}{r}\,a^{n - r}\,b^r.$ Each coefficient $\binom{n}{r}$ counts the number of ways to choose which $r$ of the $n$ factors contribute the $b$ , the same combinations you learned in Chapter 6.

The proof connects two threads of the course: algebra (multiplying $n$ copies) and combinatorics (counting choices). Every later use of binomial coefficients , in probability, in series, in derivatives of polynomials , descends from this single identity.

In Class XII you will extend the theorem to any real exponent (using calculus). For now we focus on integer exponents and master:

• writing the expansion;
• finding any specific term (the general term);
• identifying the middle term(s);
• using the expansion to compute values and prove identities;
• properties of binomial coefficients like $\binom{n}{r} = \binom{n}{n - r}$ and $\sum_r \binom{n}{r} = 2^n$.

What's inside

1. Binomial theorem for positive integral indices , statement and proof.
2. General term and any specific term , find $T_{r+1}$.
3. Middle term(s) and term independent of $x$.
4. Properties and identities of binomial coefficients.
5. Applications , divisibility, numerical estimates, identities.
6. Multinomial preview and miscellaneous problems.

Key results / Formula card

How to read this chapter

For any problem, the first step is to identify $n$, $a$, $b$. The second is to write $T_{r+1}$ and pick the right $r$. Keep Pascal's triangle in your notebook for the first few $n$ and you can verify any expansion at a glance.