For the set $X = \{ 1,2,3,4 \}$, how many functions $f : X \rightarrow X$ satisfy the following condition? [3 points] $\square$
(1) 64
(2) 68
(3) 72
(4) 76
(5) 80

Let $A = \{1, \ldots, 5\}$ and $B = \{1, \ldots, 10\}$. Then the number of ordered pairs $(f, g)$ of functions $f : A \rightarrow B$ and $g : B \rightarrow A$ satisfying $(g \circ f)(a) = a$ for all $a \in A$ is
(A) $\frac{10!}{5!} \times 5^5$
(B) $5^{10} \times 5!$
(C) $10! \times 5!$
(D) $\binom{10}{5} \times 10^5$

Let $A = \{a, b, c\}$ and $B = \{1, 2, 3, 4\}$. Then the number of elements in the set $C = \{f: A \rightarrow B \mid 2 \in f(A)$ and $f$ is not one-one$\}$ is ...

Let $A = \{ 1,2,3 , \ldots , 10 \}$ and $f : A \rightarrow A$ be defined as $$f ( k ) = \left\{ \begin{array} { c l } k + 1 & \text { if } k \text { is odd } \\ k & \text { if } k \text { is even } \end{array} \right.$$ Then the number of possible functions $g : A \rightarrow A$ such that $g o f = f$ is:
(1) ${ } ^ { 10 } \mathrm { C } _ { 5 }$
(2) $5 ^ { 5 }$
(3) 5 !
(4) $10 ^ { 5 }$

The number of functions $f$, from the set $A = \left\{ x \in N : x ^ { 2 } - 10 x + 9 \leq 0 \right\}$ to the set $B = \left\{ n ^ { 2 } : n \in N \right\}$ such that $f ( x ) \leq ( x - 3 ) ^ { 2 } + 1$, for every $x \in A$, is $\_\_\_\_$ .

The number of functions $f : \{ 1,2,3,4 \} \rightarrow \{ \mathrm { a } \in \mathbb { Z } : | \mathrm { a } | \leq 8 \}$ satisfying $f ( \mathrm { n } ) + \frac { 1 } { \mathrm { n } } f ( \mathrm { n } + 1 ) = 1 , \forall \mathrm { n } \in \{ 1,2,3 \}$ is
(1) 3
(2) 4
(3) 1
(4) 2

Let $A = \{1, 2, 3, 5, 8, 9\}$. Then the number of possible functions $f: A \rightarrow A$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to

Let $\mathrm { A } = \{ 1,2,3,4 \}$ and $\mathrm { B } = \{ 1,4,9,16 \}$. Then the number of many-one functions $f : \mathrm { A } \rightarrow \mathrm { B }$ such that $1 \in f ( \mathrm {~A} )$ is equal to :
(1) 151
(2) 139
(3) 163
(4) 127

Let $A = \{ 1,2,3 \}$ and $f : A \rightarrow A$ be a function. How many one-to-one functions $f$ satisfy the condition
$$f ( n ) \neq n$$
for every $n \in A$?
A) 1
B) 2
C) 3
D) 4
E) 5