The Binomial Theorem

Expanding (x + y)⁷ by repeated multiplication is slow and error-prone. The Binomial Theorem writes it out directly.

Pascal’s triangle

Each row begins and ends with 1, and every other entry is the sum of the two above it. Row n holds the coefficients of (x + y)ⁿ.

The coefficients are combinations

The entries of row n are nC0, nC1, up to nCn. Choosing which factors contribute a y is literally a combination.

The exponents

In (x + y)ⁿ the power of x falls from n to 0 while the power of y rises from 0 to n. Every term’s exponents total n.

One term at a time

The term containing yʳ is nCr · x^(n−r) · yʳ. That finds a single term without expanding anything else.

Subtraction

For (x − y)ⁿ the signs alternate, because each y carries a minus with it.

Expanding a power without multiplying out

(x + y)ⁿ expands into n + 1 terms whose coefficients are the binomial coefficients. Rather than multiplying n brackets together, the theorem writes the answer directly.

The coefficients are combinations

The coefficient of xᵏyⁿ⁻ᵏ is n choose k. This is because expanding the product means choosing x from k of the n brackets, and the number of such choices is exactly a combination.

Pascal's triangle

Each entry is the sum of the two above it, and row n gives the coefficients for (x + y)ⁿ. It is quick for small n and the additive rule reflects a genuine identity about combinations.

Finding one term without the rest

The general term formula lets you extract, say, the x⁵ coefficient of a twelfth power without expanding anything else. That selective extraction is what makes the theorem practically useful.