Binomial Theorem – Notes

Binomial Theorem for Positive Integral Indices

The number of terms in an expansion is one more than the power of the binomial.

The coefficients of the first and the last terms of an expansion are both 1.

(x + y)ⁿ = ⁿC₀xⁿ + ⁿc₁x^{n – 1}y + ⁿC₂x^{n – 2}y² +…. + ⁿC_n – ₁xy^{n – 1} + ⁿC_nyⁿ

ⁿC₀, ⁿC₁, ⁿC₂,… ⁿC_{n – 1}, ⁿC_n are called binomial coefficients.

General and Middle Terms

In the expansion (x + y)ⁿ

• General term, T_{r + 1} = ⁿC_rx^{n – r} y^r
• Middle term when n is even = ( ^th term