A Shortcut for Numbers Too Big to Multiply Binomial Theorem

Computing 98 to the power of 5 by directly multiplying 98 by itself five times is genuinely tedious -- but rewriting 98 as (100 minus 2) and expanding that binomial instead turns the same calculation into a short sum of round, manageable terms, each built from a power of 100 and a power of 2. This is exactly the practical motivation behind the Binomial Theorem: any expression of the form (a+b) raised to a whole-number power n can be expanded into a fixed, predictable sum of n+1 terms, without ever multiplying the original expression by itself n times directly.

Expanding (a+b) raised to increasing powers (0, 1, 2, 3, 4) and writing down just the coefficients reveals Pascal's Triangle -- each row built from the row above it by adding adjacent pairs, and each entry in that triangle is exactly a combination value nCr from the previous chapter's own formula. This connection means the Binomial Theorem's full coefficients, for any power n at all, are already known in advance: they're simply nC0, nC1, nC2, all the way through nCn, and the whole theorem can be proved rigorously using mathematical induction rather than just observed as a pattern.

Setting b to negative y instead of positive y gives (a-y)^n, whose terms simply alternate in sign. Setting a=1 and b=x gives (1+x)^n, and substituting x=1 into that expansion reveals a striking fact: the sum of ALL the binomial coefficients nC0 through nCn equals exactly 2 to the power n. Setting x=-1 into the very same expansion instead reveals that the alternating sum of those same coefficients (plus, minus, plus, minus...) always comes out to exactly 0, for any n greater than 0 -- two genuinely useful identities that fall directly out of the one general theorem, just by choosing convenient values to substitute.

Comparing (1.01) raised to the power of 1,000,000 against the plain number 10,000 seems to demand computing an impossibly huge expansion -- but using only the FIRST two terms of the full binomial expansion (which are already enough to establish the number is far larger than 10,000) settles the comparison completely, without ever needing the remaining 999,999 terms at all. A similarly elegant use of the theorem proves that 6 to the power of n, minus 5n, always leaves a remainder of exactly 1 when divided by 25 -- reached by expanding (1+5)^n and noticing every term beyond the first two is automatically divisible by 25.

Ancient Indian mathematicians already knew binomial coefficients up to n=7, and Pingala's Chhanda Shastra (around 200 BCE) arranged them into a triangle he called 'Meru-Prastara' -- the very same triangle later credited to Blaise Pascal, who only constructed his version in 1665, roughly 1800 years afterward. The same triangle also appears independently in Chinese mathematician Zhu Shijie's work from 1303, and the term 'binomial coefficient' itself is credited to the German mathematician Michael Stifel, around 1544.