Proof relating to 2-designs. Show $\lambda \le \dbinom{v-2}{k-2}$.

Question

I am required to show the following for any $2 - (v, k, \lambda)$ design:

$$\lambda \le \dbinom{v-2}{k-2}$$

and that if equal, then the design is trivial.

It's the proof I am struggling with, the second part I found trivial.

Answer 1

Hint: take the collection of all $k$-subsets, which are not included in your design. After that show that this new collection forms a new design which has parameter $\lambda^{\prime}={v-2 \choose k-2}-\lambda$. Clearly, $\lambda^{\prime}$ is non-negative, therefore $\lambda\leq {v-2 \choose k-2}$.