Let f : Z10 → Z10 be a random permutation. What is the probability that f(0) = 0 and f(1) = 1? Express your answer as a reduced fraction without any spaces (eg, 1/3and not 12/36), or as O or 1 if appropriate.

Answered: Let f : Z10 → Z10 be a random…

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Let f : Z10 → Z10 be a random permutation. What is the probability that f(0) = 0 and f(1) = 1? Express your answer as a reduced fraction without any spaces (eg, 1/3
and not 12/36), or as O or 1 if appropriate.

Expert Solution

Step 1

Let $f : ℤ_{10} \to ℤ_{10}$ be a random permutation.

Suppose A be the event that $f (0) = 0$

B be the event that $f (1) = 1$

To find probability that $f (0) = 0$ and $f (1) = 1$ i.e. $P (A \cap B)$

Total number of elements in $ℤ_{10}$ is 10. So, number of permutations is 10!.

Therefore, total number of points in sample space is 10!

Now, when $f (0) = 0$ and $f (1) = 1$ , in that case 8 elements are there remaining in $ℤ_{10}$ for which number of permutations is 8!.

Therefore, number of points in favor of the event $(A \cap B)$ is 8!

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Let f : Z10 → Z10 be a random permutation. What is the probability that f (0) = 0 and f(1) = 1? Express your answer as a reduced fraction without any spaces (eg, 1/3 and not 12/36), or as O or 1 if appropriate.