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

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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
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up vote 2 down vote accepted

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
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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
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