For a natural number $n$ denote by $\operatorname{Map}(n)$ the set of all functions $f : \{1,2,\ldots,n\} \rightarrow \{1,2,\ldots,n\}$. For $f, g \in \operatorname{Map}(n)$, $f \circ g$ denotes the function in $\operatorname{Map}(n)$ that sends $x$ to $f(g(x))$.
(a) Let $f \in \operatorname{Map}(n)$. If for all $x \in \{1,\ldots,n\}$ $f(x) \neq x$, show that $f \circ f \neq f$.
(b) Count the number of functions $f \in \operatorname{Map}(n)$ such that $f \circ f = f$.

Let $N = \{1,2,3,4,5,6,7,8,9\}$ and $L = \{a,b,c\}$.
Statements
(29) Suppose we arrange the 12 elements of $L \cup N$ in a line such that all three letters appear consecutively (in any order). The number of such arrangements is less than $10! \times 5$. (30) More than half of the functions from $N$ to $L$ have the element $b$ in their range. (31) The number of one-to-one functions from $L$ to $N$ is less than 512. (32) The number of functions from $N$ to $L$ that do not map consecutive numbers to consecutive letters is greater than 512. (e.g., $f(1) = b$ and $f(2) = a$ or $c$ is not allowed. $f(1) = a$ and $f(2) = c$ is allowed. So is $f(1) = f(2)$.)

For a natural number $n$, $f ( n )$ is defined as follows:
$$f ( n ) = \begin{cases} \log _ { 3 } n & ( n \text{ is odd} ) \\ \log _ { 2 } n & ( n \text{ is even} ) \end{cases}$$
For two natural numbers $m , n$ with $m, n \leq 20$, how many ordered pairs $( m , n )$ satisfy $f ( m n ) = f ( m ) + f ( n )$?
(1) 220
(2) 230
(3) 240
(4) 250
(5) 260

For the set $X = \{ x \mid x \text{ is a natural number not exceeding } 10 \}$, find the number of functions $f : X \rightarrow X$ satisfying the following conditions. [4 points] (가) For all natural numbers $x$ not exceeding 9, $f ( x ) \leq f ( x + 1 )$. (나) When $1 \leq x \leq 5$, $f ( x ) \leq x$, and when $6 \leq x \leq 10$, $f ( x ) \geq x$. (다) $f ( 6 ) = f ( 5 ) + 6$

For the set $X = \{1, 2, 3, 4, 5, 6\}$, how many functions $f : X \rightarrow X$ satisfy the following conditions? [4 points] (가) The value of $f(1) \times f(6)$ is a divisor of 6. (나) $2f(1) \leq f(2) \leq f(3) \leq f(4) \leq f(5) \leq 2f(6)$
(1) 166
(2) 171
(3) 176
(4) 181
(5) 186

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
What is $S(p, n)$ for $p < n$?

For every pair $(p, k)$ of nonzero natural numbers, we denote by $S(p, k)$ the number of surjections from $\llbracket 1, p \rrbracket$ to $\llbracket 1, k \rrbracket$. Consistently, for every $p \in \mathbb{N}^*$, we set $S(p, 0) = 0$.
Determine $S(n, n)$.

Let $A$ be the set $\{ 1,2 , \ldots , 6 \}$. How many functions from $A$ to $A$ are there such that the range of $f$ has exactly 5 elements?
(a) 240
(b) 720
(c) 1800
(d) 10800

Let the set of all relations $R$ on the set $\{ a , b , c , d , e , f \}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by $\mathcal { S }$.
Then the number of elements in $\mathcal { S }$ is $\_\_\_\_$ .

Let $x$ denote the total number of one-one functions from a set $A$ with 3 elements to a set $B$ with 5 elements and $y$ denote the total number of one-one functions from the set $A$ to the set $A \times B$. Then:
(1) $y = 273 x$
(2) $2 y = 273 x$
(3) $2 y = 91 x$
(4) $y = 91 x$

The total number of functions, $f : \{1,2,3,4\} \rightarrow \{1,2,3,4,5,6\}$ such that $f(1) + f(2) = f(3)$, is equal to
(1) 60
(2) 90
(3) 108
(4) 126

Let $A = \{2, 3, 4\}$ and $B = \{8, 9, 12\}$. Then the number of elements in the relation $R = \{ ( ( a _ { 1 } , b _ { 1 } ) , ( a _ { 2 } , b _ { 2 } ) ) \in ( A \times B ) \times ( A \times B ) : a _ { 1 }$ divides $b _ { 2 }$ and $a _ { 2 }$ divides $b _ { 1 } \}$ is
(1) 36
(2) 24
(3) 18
(4) 12

Let $[ t ]$ be the greatest integer less than or equal to $t$. Let $A$ be the set of all prime factors of 2310 and $f : A \rightarrow \mathbb { Z }$ be the function $f ( x ) = \left[ \log _ { 2 } \left( x ^ { 2 } + \left[ \frac { x ^ { 3 } } { 5 } \right] \right) \right]$. The number of one-to-one functions from $A$ to the range of $f$ is
(1) 25
(2) 24
(3) 20
(4) 120

Number of functions $f : \{1, 2, \ldots, 100\} \rightarrow \{0, 1\}$, that assign 1 to exactly one of the positive integers less than or equal to 98, is equal to $\_\_\_\_$.