Binomial Theorem

Multinomial Theorem and Multiplication

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Multinomial Theorem and Multiplication

Multinomial Theorem and Multiplication

This lesson covers the following key concepts:

  • Multinomial theorem for (x₁ + x₂ + ... + xₖ)ⁿ
  • Finding coefficient in multinomial expansion
  • Multiplication of binomial expansions
  • Product of two binomials
  • Applications to finding specific coefficients
  • Trinomial expansion

Important Formulas

  • (x1+x2+...+xk)n=∑n!r1!r2!...rk!x1r1x2r2...xkrk(x_1 + x_2 + ... + x_k)^n = \sum \frac{n!}{r_1! r_2! ... r_k!} x_1^{r_1} x_2^{r_2} ... x_k^{r_k}
  • where r1+r2+...+rk=n\text{where } r_1 + r_2 + ... + r_k = n
  • (1+x)m(1+x)n=(1+x)m+n(1 + x)^m (1 + x)^n = (1 + x)^{m+n}
  • Coefficient of xr in (1+x)m(1+x)n=∑k=0rmCk⋅nCr−k\text{Coefficient of } x^r \text{ in } (1+x)^m(1+x)^n = \sum_{k=0}^{r} {}^mC_k \cdot {}^nC_{r-k}