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?

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Should the $?$ be $=$s? –  Pedro Tamaroff Nov 7 '12 at 0:05

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?

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Beautifully so. Thank you. –  Doug Smith Nov 7 '12 at 0:10
Ellipses often lead to confusion around terms that look like $n - k + 1$; when in doubt, try to write in an extra term on either side of the ellipses. This often (but definitely not always) clarifies things. –  Benjamin Dickman Nov 7 '12 at 0:14