Mistakes Give Rise To Problems-19

Algebra Level 4

We know the following property of exponents that:

\[\left(\dfrac{a}{b}\right)^{c}=\dfrac{a^{c}}{b^{c}},\]

but we can't do this in factorials as:

\[\left(\dfrac{a}{b}\right)!=\dfrac{a!}{b!}\]

In general, doing this will be a Big Mistake. However, this holds for certain integers \(a\) and \(b.\) For how many ordered pairs of integers \((a,b)\) where \(1≤b≤a≤100\) does this hold?

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