A first-year general education student chooses three specializations from the twelve offered. The number of possible combinations is: a. 1728 b. 1320 c. 220 d. 33

For each of the following statements, indicate whether it is true or false. Each answer must be justified. An unjustified answer earns no points.

1. Two football teams of 22 and 25 players shake hands at the end of a match. Each player from one team shakes hands once with each player from the other team.

Statement 1 47 handshakes were exchanged.
2. A race involves 18 competitors. The three first-place finishers are rewarded indiscriminately by offering the same prize to each.
Statement 2 There are 4896 possibilities for distributing these prizes.
3. An association organizes a hurdle race competition that will establish a podium (the podium consists of the three best athletes ranked in their order of arrival). Seven athletes participate in the tournament. Jacques is one of them.
Statement 3 There are 90 different podiums on which Jacques appears.
4. Let $X _ { 1 }$ and $X _ { 2 }$ be two random variables with the same distribution given by the table below:

$x _ { i }$	- 2	- 1	2	5
$P \left( X = x _ { i } \right)$	0.1	0.4	0.3	0.2

We assume that $X _ { 1 }$ and $X _ { 2 }$ are independent and we consider $Y$ the random variable sum of these two random variables. Statement 4 $P ( Y = 4 ) = 0.25$.
5. A swimmer trains with the objective of swimming 50 metres freestyle in less than 25 seconds. Through training, it turns out that the probability of achieving this is 0.85. He performs, on one day, 20 timed 50-metre swims. We denote by $X$ the random variable that counts the number of times he swims this distance in less than 25 seconds on this day. We admit that $X$ follows the binomial distribution with parameters $n = 20$ and $p = 0.85$.
Statement 5 Given that he achieved his objective at least 15 times, an approximate value to $10 ^ { - 3 }$ of the probability that he achieved it at least 18 times is 0.434.

O número de combinações de 8 elementos tomados 3 a 3 é
(A) 24 (B) 40 (C) 56 (D) 112 (E) 336

QUESTION 165
The number of combinations of 6 elements taken 2 at a time is
(A) 12
(B) 15
(C) 18
(D) 21
(E) 24

QUESTION 172
The value of $\binom{5}{2}$ is
(A) 5
(B) 8
(C) 10
(D) 12
(E) 15

Not being fans of practicing sports, a group of friends decided to hold a soccer tournament using a video game. They decided that each player plays only once against each of the other players. The champion will be the one who gets the highest number of points. They observed that the number of matches played depends on the number of players, as shown in the table:

\begin{tabular}{ c } Number of

players

& 2 & 3 & 4 & 5 & 6 & 7 \hline

Number of

matches

& 1 & 3 & 6 & 10 & 15 & 21 \hline \end{tabular}
If the number of players is 8, how many matches will be played?
(A) 64
(B) 56
(C) 49
(D) 36
(E) 28

A committee of 3 people is to be chosen from a group of 7. How many different committees are possible?
(A) 21
(B) 28
(C) 35
(D) 42
(E) 56

The value of $\binom{6}{2}$ is:
(A) 10
(B) 12
(C) 15
(D) 18
(E) 20

For a natural number $r$, when ${}_{3}\mathrm{H}_{r} = {}_{7}\mathrm{C}_{2}$, find the value of ${}_{5}\mathrm{H}_{r}$. [3 points]

Find the value of ${}_{4}\mathrm{H}_{2}$. [3 points]

Find the value of ${}_{5}\mathrm{C}_{3}$. [3 points]

Find the value of ${ } _ { 5 } \mathrm { C } _ { 3 }$. [3 points]

Let $$\begin{gathered} S _ { 1 } = \{ ( i , j , k ) : i , j , k \in \{ 1,2 , \ldots , 10 \} \} , \\ S _ { 2 } = \{ ( i , j ) : 1 \leq i < j + 2 \leq 10 , i , j \in \{ 1,2 , \ldots , 10 \} \} , \\ S _ { 3 } = \{ ( i , j , k , l ) : 1 \leq i < j < k < l , \quad i , j , k , l \in \{ 1,2 , \ldots , 10 \} \} \end{gathered}$$ and $$S _ { 4 } = \{ ( i , j , k , l ) : i , j , k \text { and } l \text { are distinct elements in } \{ 1,2 , \ldots , 10 \} \} .$$ If the total number of elements in the set $S _ { r }$ is $n _ { r } , r = 1,2,3,4$, then which of the following statements is (are) TRUE ?
(A) $n _ { 1 } = 1000$
(B) $n _ { 2 } = 44$
(C) $n _ { 3 } = 220$
(D) $\frac { n _ { 4 } } { 12 } = 420$

If the number of 5-element subsets of the set $A = \left\{ a _ { 1 } , a _ { 2 } , \ldots , a _ { 20 } \right\}$ of 20 distinct elements is $k$ times the number of 5-element subsets containing $a _ { 4 }$, then $k$ is
(1) 5
(2) $\frac { 20 } { 7 }$
(3) 4
(4) $\frac { 10 } { 3 }$

If $\frac { { } ^ { n + 2 } C _ { 6 } } { { } ^ { n - 2 } P _ { 2 } } = 11$, then $n$ satisfies the equation:
(1) $n ^ { 2 } + n - 110 = 0$
(2) $n ^ { 2 } + 2 n - 80 = 0$
(3) $n ^ { 2 } + 3 n - 108 = 0$
(4) $n ^ { 2 } + 5 n - 84 = 0$

Consider three boxes, each containing 10 balls labelled $1,2 , \ldots , 10$. Suppose one ball is randomly drawn from each of the boxes. Denote by $n _ { i }$, the label of the ball drawn from the $i ^ { \text {th } }$ 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:
(1) 240
(2) 82
(3) 120
(4) 164

The number of ordered pairs $( r , k )$ for which $6 . { } ^ { 35 } C _ { r } = \left( k ^ { 2 } - 3 \right) . { } ^ { 36 } C _ { r + 1 }$, where $k$ is an integer is
(1) 3
(2) 2
(3) 6
(4) 4

If $a , b$ and $c$ are the greatest values of ${}^{ 19 } C _ { p } , {}^{ 20 } C _ { q }$ and ${}^{ 21 } C _ { r }$ respectively, then:
(1) $\frac { a } { 11 } = \frac { b } { 22 } = \frac { c } { 21 }$
(2) $\frac { a } { 10 } = \frac { b } { 20 } = \frac { c } { 21 }$
(3) $\frac { a } { 11 } = \frac { b } { 22 } = \frac { c } { 42 }$
(4) $\frac { a } { 10 } = \frac { b } { 11 } = \frac { c } { 42 }$

If all the six digit numbers $\mathrm { x } _ { 1 } \mathrm { x } _ { 2 } \mathrm { x } _ { 3 } \mathrm { x } _ { 4 } \mathrm { x } _ { 5 } \mathrm { x } _ { 6 }$ with $0 < \mathrm { x } _ { 1 } < \mathrm { x } _ { 2 } < \mathrm { x } _ { 3 } < \mathrm { x } _ { 4 } < \mathrm { x } _ { 5 } < \mathrm { x } _ { 6 }$ are arranged in the increasing order, then the sum of the digits in the $72 ^ { \text {th} }$ number is $\_\_\_\_$ .

Five digit numbers are formed using the digits $1,2,3,5,7$ with repetitions and are written in descending order with serial numbers. For example, the number 77777 has serial number 1. Then the serial number of 35337 is

If for some $m, n$; ${}^{6}C_m + 2\,{}^{6}C_{m+1} + {}^{6}C_{m+2} > {}^{8}C_3$ and ${}^{n-1}P_3 : {}^{n}P_4 = 1 : 8$, then ${}^{n}P_{m+1} + {}^{n+1}C_m$ is equal to
(1) 380
(2) 376
(3) 384
(4) 372

Let ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } - 1 } = 28 , { } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } } = 56$ and ${ } ^ { \mathrm { n } } \mathrm { C } _ { \mathrm { r } + 1 } = 70$. Let $\mathrm { A } ( 4 \cos t , 4 \sin t ) , \mathrm { B } ( 2 \sin t , - 2 \cos \mathrm { t } )$ and $C \left( 3 r - n , r ^ { 2 } - n - 1 \right)$ be the vertices of a triangle $ABC$, where $t$ is a parameter. If $( 3 x - 1 ) ^ { 2 } + ( 3 y ) ^ { 2 } = \alpha$, is the locus of the centroid of triangle ABC, then $\alpha$ equals
(1) 6
(2) 18
(3) 8
(4) 20

An ice cream shop needs to prepare at least $n$ buckets of different flavors of ice cream to satisfy the advertisement claim that ``the number of combinations of selecting two scoops of different flavors exceeds 100 types.'' How many ways can a customer select two scoops (which may be the same flavor) from $n$ buckets?
(1) 101
(2) 105
(3) 115
(4) 120
(5) 225

Let $a , b , c$ be real numbers and $0 < b < 1$ such that
$$\begin{aligned} & a = b \cdot c \\ & a + c = b \end{aligned}$$
Given this, which of the following orderings is correct?
A) $a < b < c$
B) $a < c < b$
C) $b < a < c$
D) $c < a < b$
E) $c < b < a$

Let $a$ and $b$ be digits. Given the sets
$$\begin{aligned} & A = \{ 5,6,7,8,9 \} \\ & B = \{ 1,4,5,7 \} \\ & C = \{ a , b \} \end{aligned}$$
If the number of elements in the Cartesian product $(A \cup C) \times (B \cup C)$ is 28, what is the sum $a + b$?
A) 5
B) 6
C) 8
D) 9
E) 11