# Addition Rule for Gaussian Binomial Coefficients

## Theorem

Let $\dbinom n m_q$ denote a Gaussian binomial coefficient.

### Formulation 1

$\dbinom n m_q = \dbinom {n - 1} m_q + \dbinom {n - 1} {m - 1}_q q^{n - m}$

### Formulation 2

$\dbinom n m_q = \dbinom {n - 1} m_q q^m + \dbinom {n - 1} {m - 1}_q$