Permutations and Combinations 2 Question 16
16. Let $A$ be a set of $n$ distinct elements. Then, the total number of distinct functions from $A$ to $A$ is… and out of these… are onto functions.
$(1985,2 M)$
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Answer:
Correct Answer: 16. $n^{n}, \sum _{r=1}^{n}(-1)^{n-r} C _r(r)^{n}$
Solution:
- Let $A={x _1, x _2, \ldots, x _n }$
$\therefore$ Number of functions from $A$ to $A$ is $n^{n}$ and out of these $\sum _{r=1}^{n}(-1)^{n-r}{ }^{n} C _r(r)^{n}$ are onto functions.
