Chapter 7 ยท Introduction

Chapter 7 โ€” Binomial Theorem

Multiply $(a+b)$ by itself again and again and a triangle of numbers appears โ€” the same numbers ${}^nC_r$ that count selections. The binomial theorem turns that coincidence into a tool for exact arithmetic, quick estimates and divisibility proofs.

More than two thousand years ago the Indian scholar Pingala studied the rhythms of Sanskrit verse, built from short and long syllables. How many rhythms of four syllables contain exactly two long ones? Later commentators arranged the answers for every length in a triangular table they called the Meru-prastฤra, the "staircase of Mount Meru": $1$; $1\ 1$; $1\ 2\ 1$; $1\ 3\ 3\ 1$; $1\ 4\ 6\ 4\ 1$; โ€ฆ Each number is the sum of the two above it. The same triangle turns up in Persian, Chinese and European mathematics, and today it carries the name of Blaise Pascal, who wrote a famous treatise on it in 1653.

The surprise is that this counting table is also a multiplication table. Expand $(a+b)^4$ and its coefficients are $1,4,6,4,1$ โ€” the row for rhythms of four syllables. The reason is a counting argument: multiplying out $n$ brackets means choosing $a$ or $b$ from each one, exactly as a rhythm chooses short or long for each syllable. In this chapter you will build the triangle, prove the binomial theorem $(a+b)^n=\sum{}^nC_r\,a^{n-r}b^r$ by counting, and then use it โ€” to compute $99^5$ in your head, to estimate $(1.01)^{10}$ to four decimal places, to decide whether $(1.01)^{1000}$ is bigger than $1000$, and to prove that $9^n-8n-1$ is always a multiple of $64$. In Class 12 and in JEE, the same coefficients reappear in probability and in calculus.

Build: Pascal's triangle by additionโ†’Count: the entries are ${}^nC_r$โ†’Prove: $(a+b)^n=\sum{}^nC_r\,a^{n-r}b^r$โ†’Use: exact powers, estimates, remainders
๐ŸŽฏ By the end of this chapter you canโ€ฆ
  • build any row of Pascal's triangle and explain why each entry is the sum of the two above it;
  • state the binomial theorem for a positive integral index and prove it by counting;
  • expand powers such as $(2x-3y)^4$, $\left(x^2+\tfrac2x\right)^4$, $(1+x)^n$ and $(1-x)^n$, and use ${}^nC_0+{}^nC_1+\cdots+{}^nC_n=2^n$;
  • evaluate powers such as $99^5$ or $(10.2)^4$ exactly, and approximate powers such as $(0.98)^5$ to a required accuracy;
  • compare large powers with the bound $(1+x)^n\ge1+nx$, and prove divisibility results and find remainders with the binomial expansion.
Teaser โ€” The Bean Machine

Balls fall through $n$ rows of pins; at every pin a ball bounces left or right with equal chance, and it lands in one of $n+1$ bins. A bin is reached by every path with the same number of right-bounces โ€” and the number of such paths is an entry of row $n$ of Pascal's triangle. Watch the bins fill up and compare them with the purple outline, which shows $\tfrac{\text{balls}}{2^n}\times{}^nC_r$. The more balls, the closer the match.

6
balls droppedโ€”
bin countsโ€”
row $n$ (paths to each bin)โ€”

Sections in this chapter

7.1

Pascal's Triangle and the Binomial Theorem

The triangle and its addition rule, why its entries are ${}^nC_r$, the binomial theorem with its counting proof, and the special cases $(a-b)^n$, $(1\pm x)^n$ and the row sum $2^n$.

$(a+b)^n=\sum_{r=0}^{n}{}^nC_r\,a^{n-r}b^r$
Open section โ†’7.2

Applications of the Binomial Expansion

Evaluating powers such as $99^5$ exactly, approximating $(1.01)^{10}$, comparing $(1.01)^{1000}$ with $1000$, and divisibility and remainder problems.

$(1+x)^n\ge1+nx\ \ (x\ge0)$
Open section โ†’
โœ… Before you start
You need the factorial $n!$ from Section 6.2 and, above all, the combinations ${}^nC_r=\dfrac{n!}{r!\,(n-r)!}$ of Section 6.3, with their two key properties ${}^nC_r={}^nC_{n-r}$ and ${}^nC_{r-1}+{}^nC_r={}^{n+1}C_r$; the fundamental principle of counting from Section 6.1 drives the proof of the binomial theorem. From Class 9 and Class 10 Mathematics you need the identities $(a\pm b)^2$ and $(a\pm b)^3$, the laws of indices, and confident work with surds and decimals. One exercise uses $i^2=-1$ from Section 4.1, and the applications use percentages and compound growth.
โ† 6.3 Combinations7.1 Pascal's Triangle and the Binomial Theorem โ†’