Let $S = \{ 1, 2, \ldots , n \}$. The number of possible pairs of the form $(A, B)$ with $A \subseteq B$ for subsets $A$ and $B$ of $S$ is
(A) $2 ^ { n }$
(B) $3 ^ { n }$
(C) $\sum _ { k = 0 } ^ { n } \binom { n } { k } \binom { n } { n - k }$
(D) $n !$

The number of maps $f$ from the set $\{ 1, 2, 3 \}$ into the set $\{ 1, 2, 3, 4, 5 \}$ such that $f(i) \leq f(j)$ whenever $i < j$ is
(A) 60
(B) 50
(C) 35
(D) 30

The number of maps $f$ from the set $\{ 1, 2, 3 \}$ into the set $\{ 1, 2, 3, 4, 5 \}$ such that $f ( i ) \leq f ( j )$ whenever $i < j$ is
(A) 60
(B) 50
(C) 35
(D) 30

Consider three boxes, each containing 10 balls labelled $1, 2, \ldots, 10$. Suppose one ball is drawn from each of the boxes. Denote by $n_i$, the label of the ball drawn from the $i$-th box, $i = 1, 2, 3$. Then the number of ways in which the balls can be chosen such that $n_1 < n_2 < n_3$ is
(A) 120
(B) 130
(C) 150
(D) 160

Consider three boxes, each containing 10 balls labelled $1, 2, \ldots, 10$. Suppose one ball is drawn from each of the boxes. Denote by $n _ { i }$, the label of the ball drawn from the $i$-th box, $i = 1, 2, 3$. Then the number of ways in which the balls can be chosen such that $n _ { 1 } < n _ { 2 } < n _ { 3 }$ is
(A) 120
(B) 130
(C) 150
(D) 160

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom{10}{3} \binom{7}{4} 2^4$
(B) $\binom{10}{3} \binom{7}{4}$
(C) $\binom{10}{3} 2^7$
(D) $\binom{10}{3} \binom{14}{4}$

A box contains 10 red cards numbered $1, \ldots, 10$ and 10 black cards numbered $1, \ldots, 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(A) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$

The number of ways in which one can select six distinct integers from the set $\{1, 2, 3, \cdots, 49\}$, such that no two consecutive integers are selected, is
(A) $\binom{49}{6} - 5\binom{48}{5}$
(B) $\binom{43}{6}$
(C) $\binom{25}{6}$
(D) $\binom{44}{6}$

The number of ways in which one can select six distinct integers from the set $\{ 1, 2, 3, \cdots, 49 \}$, such that no two consecutive integers are selected, is
(A) $\binom { 49 } { 6 } - 5 \binom { 48 } { 5 }$
(B) $\binom { 43 } { 6 }$
(C) $\binom { 25 } { 6 }$
(D) $\binom { 44 } { 6 }$

Let $V$ be the set of vertices of a regular polygon with twenty sides. Three distinct vertices are chosen at random from $V$. Then, the probability that the chosen triplet are the vertices of a right angled triangle is
(A) $\frac{7}{19}$
(B) $\frac{3}{19}$
(C) $\frac{3}{38}$
(D) $\frac{1}{38}$.

An office has 8 officers including two who are twins. Two teams, Red and Blue, of 4 officers each are to be formed randomly. What is the probability that the twins would be together in the Red team?
(A) $\frac { 1 } { 6 }$
(B) $\frac { 3 } { 7 }$
(C) $\frac { 1 } { 4 }$
(D) $\frac { 3 } { 14 }$

An up-right path is a sequence of points $\mathbf { a } _ { 0 } = \left( x _ { 0 } , y _ { 0 } \right) , \mathbf { a } _ { 1 } = \left( x _ { 1 } , y _ { 1 } \right) , \mathbf { a } _ { 2 } = ( x _ { 2 } , y _ { 2 } ), \ldots$ such that $\mathbf { a } _ { i + 1 } - \mathbf { a } _ { i }$ is either $( 1,0 )$ or $( 0,1 )$. The number of up-right paths from $( 0,0 )$ to $( 100,100 )$ which pass through $( 1,2 )$ is:
(A) $3 \cdot \binom { 197 } { 99 }$
(B) $3 \cdot \binom { 100 } { 50 }$
(C) $2 \cdot \binom { 197 } { 98 }$
(D) $3 \cdot \binom { 197 } { 100 }$.

A school allowed the students of a class to go to swim during the days March 11th to March 15, 2019. The minimum number of students the class should have had that ensures that at least two of them went to swim on the same set of dates is:
(A) 6
(B) 32
(C) 33
(D) 121 .

The number of subsets of $\{ 1,2,3 , \ldots , 10 \}$ having an odd number of elements is
(A) 1024
(B) 512
(C) 256
(D) 50 .

A group of 64 players in a chess tournament needs to be divided into 32 groups of 2 players each. In how many ways can this be done?
(A) $\frac { 64 ! } { 32 ! 2 ^ { 32 } }$
(B) $\binom { 64 } { 2 } \binom { 62 } { 2 } \cdots \binom { 4 } { 2 } \binom { 2 } { 2 }$
(C) $\frac { 64 ! } { 32 ! 32 ! }$
(D) $\frac { 64 ! } { 2 ^ { 64 } }$

Let $A = \left\{ x _ { 1 } , x _ { 2 } , \ldots , x _ { 50 } \right\}$ and $B = \left\{ y _ { 1 } , y _ { 2 } , \ldots , y _ { 20 } \right\}$ be two sets of real numbers. What is the total number of functions $f : A \rightarrow B$ such that $f$ is onto and $f \left( x _ { 1 } \right) \leq f \left( x _ { 2 } \right) \leq \cdots \leq f \left( x _ { 50 } \right)$ ?
(A) $\binom { 49 } { 19 }$
(B) $\binom { 49 } { 20 }$
(C) $\binom { 50 } { 19 }$
(D) $\binom { 50 } { 20 }$

A box has 13 distinct pairs of socks. Let $p _ { r }$ denote the probability of having at least one matching pair among a bunch of $r$ socks drawn at random from the box. If $r _ { 0 }$ is the maximum possible value of $r$ such that $p _ { r } < 1$, then the value of $p _ { r _ { 0 } }$ is
(A) $1 - \frac { 12 } { { } ^ { 26 } C _ { 12 } }$.
(B) $1 - \frac { 13 } { { } ^ { 26 } C _ { 13 } }$.
(C) $1 - \frac { 2 ^ { 13 } } { { } ^ { 26 } C _ { 13 } }$.
(D) $1 - \frac { 2 ^ { 12 } } { { } ^ { 26 } C _ { 12 } }$.

Let $( n _ { 1 } , n _ { 2 } , \cdots , n _ { 12 } )$ be a permutation of the numbers $1,2 , \cdots , 12$. The number of arrangements with $$n _ { 1 } > n _ { 2 } > n _ { 3 } > n _ { 4 } > n _ { 5 } > n _ { 6 }$$ and $$n _ { 6 } < n _ { 7 } < n _ { 8 } < n _ { 9 } < n _ { 10 } < n _ { 11 } < n _ { 12 }$$ equals:
(A) $\binom { 12 } { 5 }$
(B) $\binom { 12 } { 6 }$
(C) $\binom { 11 } { 6 }$
(D) $\frac { 11 ! } { 2 }$

In how many ways can we choose $a _ { 1 } < a _ { 2 } < a _ { 3 } < a _ { 4 }$ from the set $\{ 1,2 , \ldots , 30 \}$ such that $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ are in arithmetic progression?
(A) 135
(B) 145
(C) 155
(D) 165

How many functions $f : \{ 1,2 , \ldots , 10 \} \rightarrow \{ 1 , \ldots , 2000 \}$, which satisfy $$f ( i + 1 ) - f ( i ) \geq 20 , \text { for all } 1 \leq i \leq 9 ,$$ are there?
(A) $10 ! \binom { 1829 } { 10 }$
(B) $11 ! \binom { 1830 } { 11 }$
(C) $\binom { 1829 } { 10 }$
(D) $\binom { 1830 } { 11 }$

Suppose 40 distinguishable balls are to be distributed into 4 different boxes such that each box gets exactly 10 balls. Out of these 40 balls, 10 are defective and 30 are non-defective. In how many ways can the balls be distributed such that all the defective balls go to the first two boxes?
(A) $\frac{40!}{(10!)^4}$
(B) $\frac{30! \cdot 20!}{(10!)^5}$
(C) $\frac{20! \cdot 20!}{(10!)^5}$
(D) $\frac{30! \cdot 10!}{(10!)^4}$

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 $X$ be the set $\{ 1,2,3 , \ldots , 10 \}$ and $P$ the subset $\{ 1,2,3,4,5 \}$. The number of subsets $Q$ of $X$ such that $P \cap Q = \{ 3 \}$ is
(A) 1
(B) $2 ^ { 4 }$
(C) $2 ^ { 5 }$
(D) $2 ^ { 9 }$

A box contains 10 red cards numbered $1 , \ldots , 10$ and 10 black cards numbered $1 , \ldots , 10$. In how many ways can we choose 10 out of the 20 cards so that there are exactly 3 matches, where a match means a red card and a black card with the same number?
(a) $\binom { 10 } { 3 } \binom { 7 } { 4 } 2 ^ { 4 }$.
(B) $\binom { 10 } { 3 } \binom { 7 } { 4 }$.
(C) $\binom { 10 } { 3 } 2 ^ { 7 }$.
(D) $\binom { 10 } { 3 } \binom { 14 } { 4 }$.

2. In a piggy bank there are 15 coins, of which 9 are 1 euro coins and the other 6 are 2 euro coins. 6 coins are drawn simultaneously. – What is the probability that the total value of the coins drawn is exactly 10 euros? – What is the probability that the total value of the coins drawn is at most 10 euros?