Recall the cardinality of $\mathcal{S}_n$. Deduce that $R \geq 1$.

For $k \in \llbracket 0, n \rrbracket$, show that the number of permutations of $\llbracket 1, n \rrbracket$ having exactly $k$ fixed points is $\binom{n}{k} d_{n-k}$.
Deduce that $P_n\left(X_n = k\right) = \frac{d_{n-k}}{k!(n-k)!}$.

Establish that $$\operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{S}_{n} : \varepsilon(\sigma) = -1\right\}$$ and deduce the probability that a permutation of $\mathfrak{S}_{n}$ drawn uniformly at random has a prescribed signature.

For $\sigma \in \mathfrak{S}_{n}$, specify the condition on $\nu(\sigma)$ for which $\sigma \in \mathfrak{D}_{n}$. Deduce that $$\operatorname{Card}\left\{\sigma \in \mathfrak{D}_{n} : \varepsilon(\sigma) = 1\right\} = \operatorname{Card}\left\{\sigma \in \mathfrak{D}_{n} : \varepsilon(\sigma) = -1\right\} + (-1)^{n-1}(n-1).$$

154- The number of ordered triples, with non-negative integer and non-positive integer coordinates, such that the sum of every three coordinates of each set equals $10$ and each coordinate is less than $6$, is which of the following?
$$17 \ (1) \hspace{2cm} 18 \ (2) \hspace{2cm} 20 \ (3) \hspace{2cm} 21 \ (4)$$

154- Six numbered balls are randomly placed in 3 boxes. What is the probability that no box remains empty?
(1) $\dfrac{5}{14}$ (2) $\dfrac{5}{12}$ (3) $\dfrac{3}{7}$ (4) $\dfrac{7}{12}$

148. The number of surjective (onto) functions from a set with 6 elements to a set with 3 elements is which of the following?
(1) $360$ (2) $450$ (3) $480$ (4) $540$

142- How many four-digit natural numbers are divisible by 5, with non-repeating digits?
\[ (1)\quad 948 \qquad (2)\quad 952 \qquad (3)\quad 968 \qquad (4)\quad 972 \]

148 -- In a competition, 3 drivers participate for three consecutive days with 3 cars on routes A, B, and C. Each driver selects only one route and one car per day, and the scheduling is done in the form of a Latin square. In how many ways can the scheduling be done such that on the first day, no one selects car A?

2	3	1
3	1	2
1	2	3

• [(1)] $1$
• [(2)] $2$
• [(3)] $3$
• [(4)] $4$

137. The number of correct non-negative integer solutions of the equation $x_1 + x_2 + x_3 = \dfrac{10}{x_4}$ is which of the following?
(1) $60$ (2) $72$ (3) $81$ (4) $96$

125-- How many five-digit natural numbers can be written with non-repeating digits such that among those digits, one is between two even and two odd digits?
(1) $1850$ (2) $1950$ (3) $2150$ (4) $2500$

149- How many natural numbers less than $6000$ have digit sum $8$?
(4) $168$ (3) $164$ (2) $165$ (1) $155$

21 -- In how many ways can 4 ministers, each with one assistant, sit in two rows of 8 seats facing each other so that each minister sits exactly opposite their own assistant?
(1) $24$ (2) $32$ (3) $48$ (4) $64$

Let $f(n)$ be the number of ways to write a positive integer as an ordered sum of three non-negative integers, where each integer is chosen from $\{0, 1, 2, \ldots, 2n-1\}$ (i.e., using $n$ colours with values $0$ to $2n-1$). Find $f(n)$.

A person $X$ standing at a point $P$ on a flat plane starts walking. At each step, he walks exactly 1 foot in one of the directions North, South, East or West. Suppose that after 6 steps $X$ comes to the original position $P$. Then the number of distinct paths that $X$ can take is
(a) 196
(b) 256
(c) 344
(d) 400

In how many ways can 3 couples sit around a round table such that men and women alternate and none of the couples sit together?
(a) 1
(b) 2
(c) $5! / 3$
(d) None of these.

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 $f:\{1,2,3,4\} \to \{1,2,3,4\}$ be a function such that $f(i) \neq i$ for all $i$ (i.e., a derangement). Find the number of such functions.

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

Find the sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4,5$, each digit appearing at most once.

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

The sum of all distinct four digit numbers that can be formed using the digits $1,2,3,4$, and 5, each digit appearing at most once, is
(A) 399900
(B) 399960
(C) 390000
(D) 360000

Let $A = \{ 1,2 , \ldots , n \}$. For a permutation $P = ( P ( 1 ) , P ( 2 ) , \cdots , P ( n ) )$ of the elements of $A$, let $P ( 1 )$ denote the first element of $P$. Find the number of all such permutations $P$ so that for all $i , j \in A$:

• if $i < j < P ( 1 )$, then $j$ appears before $i$ in $P$; and
• if $P ( 1 ) < i < j$, then $i$ appears before $j$ in $P$.

Let $A = \{1,2,3,4,5,6\}$ and $B = \{a,b,c,d,e\}$. How many functions $f : A \rightarrow B$ are there such that for every $x \in A$, there is one and exactly one $y \in A$ with $y \neq x$ and $f(x) = f(y)$?
(A) 450
(B) 540
(C) 900
(D) 5400.

Consider all the permutations of the twenty six English letters that start with $z$. In how many of these permutations the number of letters between $z$ and $y$ is less than those between $y$ and $x$?
(A) $6 \times 23!$
(B) $6 \times 24!$
(C) $156 \times 23!$
(D) $156 \times 24!$.