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

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) $n !$

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) $\frac{1}{38}$.

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.

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) $3 \cdot \binom { 197 } { 100 }$.

Let 10 red balls and 10 white balls be arranged in a straight line such that 10 each are on either side of a central mark. The number of such symmetrical arrangements about the central mark is
(A) $\frac { 10 ! } { 5 ! 5 ! }$
(B) $10 !$
(C) $\frac { 10 ! } { 5 ! }$
(D) $2 \cdot 10 !$

Let $P$ be a regular twelve-sided polygon. The number of right-angled triangles formed by the vertices of $P$ is
(A) 60
(B) 120
(C) 160
(D) 220 .

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) $\binom { 50 } { 20 }$

Let $S = \{ 1,2 , \ldots , n \}$. For any non-empty subset $A$ of $S$, let $l ( A )$ denote the largest number in $A$. If $f ( n ) = \sum _ { A \subseteq S } l ( A )$, that is, $f ( n )$ is the sum of the numbers $l ( A )$ while $A$ ranges over all the nonempty subsets of $S$, then $f ( n )$ is
(A) $2 ^ { n } ( n + 1 )$
(B) $2 ^ { n } ( n + 1 ) - 1$
(C) $2 ^ { n } ( n - 1 )$
(D) $2 ^ { n } ( n - 1 ) + 1$.

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) $\frac { 11 ! } { 2 }$

An urn contains 30 balls out of which one is special. If 6 of these balls are taken out at random, what is the probability that the special ball is chosen?
(A) $\frac { 1 } { 30 }$
(B) $\frac { 1 } { 6 }$
(C) $\frac { 1 } { 5 }$
(D) $\frac { 1 } { 15 }$