Q:

Could anyone explain how my textbook did this simplification?

The book is talking about proving Pascal's triangle increases until the middle, until which point it decreases.

Theorem 4.2 refers to the Multiplicative formula
I just don't understand how that left side could simplify to $1$. And how do they rearrange it?

A:

The numerator of the left side is $n(n-1)\cdots (n-k+1)$.
The numerator of the right side is $n(n-1) \cdots (n-k+1)(n-k)$.
Dividing both sides by $n(n-1)\cdots (n-k+1)$, the left's numerator becomes $1$, while the right's numerator becomes $(n-k)$.
Similarly, the denominator on the left side is $k(k-1)\cdots 1$.
The denominator on the right side is $(k+1)k(k-1)\cdots 1$.
Multiplying both sides by $k(k-1)\cdots 1$, the left's denominator becomes $1$, while the right's denominator becomes $k+1$.
This results in: $1/1 = (n-k)/(k+1)$.
Does this clear things up?

