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Binomial Theorem
Introduction
Binomial Theorem for Positive Integral Indices
Practice: Binomial Expansions
Chapter Review: Mixed Practice
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Binomial TheoremIntroduction

Reading time: ~10 min

Riya's Nani taught mathematics for thirty years and still does every sum in her head. One evening Riya tries to catch her out: "Nani, what is 101^4?" Before Riya's phone has even unlocked, Nani answers: 104060401.

Her secret is a row of numbers she learnt as a girl: 1, 4, 6, 4, 1. Give each number a two-digit slot — 1 | 04 | 06 | 04 | 01 — and the answer is sitting right there.

Does the trick always work? Pick 11 or 101 as the base, drag the power, and compare the number, cut into blocks, with Nani's row for that power. Complete all three missions.

Power: ${n}

So the trick is not magic. 101 is 100 + 1, and multiplying out (100 + 1)^n gives the numbers of Nani's row, each one multiplied by a different power of 100. Every number gets its own two-digit block — until a number in the row reaches 100 and spills into the block next door. With 11 = 10 + 1 the blocks are only one digit wide, so the trick breaks as soon as the row contains a 10. Those rows of numbers are the heart of this chapter.

You already know how to square and cube binomials such as a + b and a - b, and you have used them for shortcuts like 98^2 = (100 - 2)^2 or 999^3 = (1000 - 1)^3. But for 98^5 or 101^6, multiplying again and again becomes slow and full of chances to slip. The binomial theorem does away with that work: it writes down the expansion of (a + b)^n directly. Mathematicians later stretched it to powers that are negative integers or rational numbers, but in this chapter n is always a positive whole number — a positive integral index.

Sina