in Probability recategorized ago by
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2 votes
2 votes

Let $\mathcal{F}$ be the set of all functions mapping $\{1, \ldots, n\}$ to $\{1, \ldots, m\}$. Let $f$ be a function that is chosen uniformly at random from $\mathcal{F}$. Let $x, y$ be distinct elements from the set $\{1, \ldots, n\}$. Let $p$ denote the probability that $f(x)=f(y)$. Then,

  1. $p=0$
  2. $p=\frac{1}{n^m}$
  3. $0<p \leq \frac{1}{m^n}$
  4. $p=\frac{1}{m}$
  5. $p=\frac{1}{n}$
in Probability recategorized ago by

1 Answer

3 votes
3 votes

Sample space will be Total no of functions = $m^{n}$

Since we need only the functions where f(x) = f(y) 

so f(x) can map to any of the m elements which is same as selecting one element from m elements 

Answer:

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