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Question

Answers

a. $\dfrac{1}{{{n^n}}}$

b. $\dfrac{1}{{n!}}$

c. $\dfrac{{(n - 1)!}}{{{n^{n - 1}}}}$

d. None of these.

Answer
Verified

Probability: -

Probability=Favourable outcomes/Total number of outcomes.

Probability can never be greater than one.

One to one mapping: -

One to one or (1-1) function is a relation that preserves the “uniqueness”. Every unique member of the function’s domain is mapped to the unique member of the function’s range. This mapping is sometimes also called as injective mapping.

Number of ways first element of set A can be mapped = n

Number of ways second element of set A can be mapped = n

Number of ways the third element of set A can be mapped = n and so on.

Total number of mapping from set A to itself=$n \times n \times ...... \times n = {n^n}$ (it is being multiplied to n times)

For one to one mapping: -

Number of ways to map first element in set A=n

Number of ways to map second element in set A=n-1

Number of ways to map third element in set A=n-2

Number of ways to map nth element in set A=1

Total number of one to one mapping from set A to itself=\[n \times (n - 1) \times (n - 2)..... \times 1 = n!\]

$\therefore $ Required probability=Total number of one to one mapping from set A to itself/Total number of mapping from set A to itself

\[ \Rightarrow \dfrac{{n!}}{{{n^n}}} = \dfrac{{n(n - 1)!}}{{{n^n}}}\] (Opening factorial using formula$n! = n(n - 1)(n - 2)(n - 3) \times ....... \times 3 \times 2 \times 1$ )

\[\therefore \dfrac{{(n - 1)!}}{{{n^{n - 1}}}}\] (Using identity $\dfrac{{{x^a}}}{{{x^b}}} = {x^{a - b}}$ )

Hence option C is correct.

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