inclusionAI/Ring-lite-rl-data huggingface dataset:
Given that the general term of the sequence $\{a_n\}$ is $a_n=2^{n-1}$, and the general term of the sequence $\{b_n\}$ is $b_n=3n$, let set $A=\{a_1,a_2,\ldots,a_n,\ldots\}$, $B=\{b_1,b_2,\ldots,b_n,\ldots\}$, $n\in\mathbb{N}^*$. The sequence $\{c_n\}$ is formed by arranging the elements of set $A\cup B$ in ascending order. Find the sum of the first 28 terms of the sequence $\{c_n\}$, denoted as $S_{28}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{820}
15.1. $[9.3$ (15 points)] In trapezoid $ABCD$, diagonal $AC$ is equal to 1 and is also its height. Perpendiculars $AE$ and $CF$ are drawn from points $A$ and $C$ to sides $CD$ and $AB$ respectively. Find $AD$, if $AD=CF$ and $BC=CE$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{\sqrt{2}-1}}
In a factory, a total of $100$ parts were produced. Among them, A produced $0$ parts, with $35$ being qualified. B produced $60$ parts, with $50$ being qualified. Let event $A$ be "Selecting a part from the $100$ parts at random, and the part is qualified", and event $B$ be "Selecting a part from the $100$ parts at random, and the part is produced by A". Then, the probability of $A$ given $B$ is \_\_\_\_\_\_.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{8}}
A point $Q$ is chosen in the interior of $\triangle DEF$ such that when lines are drawn through $Q$ parallel to the sides of $\triangle DEF$, the resulting smaller triangles $u_{1}$, $u_{2}$, and $u_{3}$ have areas $16$, $25$, and $36$, respectively. Furthermore, a circle centered at $Q$ inside $\triangle DEF$ cuts off a segment from $u_3$ with area $9$. Find the area of $\triangle DEF$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{225}
Determine the number of increasing sequences of positive integers $a_1 \le a_2 \le a_3 \le \cdots \le a_8 \le 1023$ such that $a_i-i$ is even for $1\le i \le 8$. The answer can be expressed as $\binom{m}{n}$ for some $m > n$. Compute the remainder when $m$ is divided by 1000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{515}
Find the mass percentage of O in Aluminum oxide.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{47.07\%}
4. To celebrate Barbara's birthday, Alberto proposes the following game: given the set of numbers $0,1, \ldots, 1024$, Barbara removes $2^{9}$ numbers from this set. In the next step, Alberto removes $2^{8}$ numbers from the remaining ones. It's Barbara's turn again, and she removes $2^{7}$ numbers from the remaining ones, and so on, until only 2 numbers $a$ and $b$ are left. At this point, Alberto must pay Barbara $|a-b|$ euros.

Determine the maximum amount of euros that Barbara is sure to collect, regardless of the strategy adopted by Alberto.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{32}
Find the expression for the optimal value of \(\alpha\) in the capacitor charging equation to maximize the slope of the voltage curve at time \(T\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{T}}
Let square $WXYZ$ have sides of length $8$. An equilateral triangle is drawn such that no point of the triangle lies outside $WXYZ$. Determine the maximum possible area of such a triangle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16\sqrt{3}}
Given a sector that is one-fourth of a circle with radius $R$. Find the length of the tangent drawn from the midpoint of the arc to its intersection with the extensions of the sector's radial boundaries.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2R}
Find the last three digits of 
\[2008^{2007^{\cdot^{\cdot^{\cdot ^{2^1}}}}}.\]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
A digit was crossed out from a six-digit number, resulting in a five-digit number. When this five-digit number was subtracted from the original six-digit number, the result was 654321. Find the original six-digit number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{727023}
A survey is conducted on the website of the football club "Rostov" to determine which of the $m$ football players visitors consider the best based on the season's results. Each visitor votes once for one football player. The website displays the rating of each football player as the percentage of votes they have received, rounded to the nearest whole number. After several visitors have voted, the total rating of the nominees added up to $95 \%$. What is the smallest possible value of $m$ for which this is possible?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Create a cube $C_{1}$ with edge length 1. Take the centers of the faces and connect them to form an octahedron $O_{1}$. Take the centers of the octahedron's faces and connect them to form a new cube $C_{2}$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{54+9 \sqrt{3}}{8}}
Compute 
$\sum_{n=3}^{\infty}{\frac{n^{2}-2}{\left(n^{2}-1\right)\left(n^{2}-4\right)}}.$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$\frac{4}{9}$}
Let $ABC$ be a triangle with $AB=20$, $AC=34$, and $BC=42$. Let $\omega_1$ and $\omega_2$ be the semicircles with diameters $\overline{AB}$ and $\overline{AC}$ erected outwards of $\triangle ABC$. Denote by $\ell$ the common external tangent to $\omega_1$ and $\omega_2$. The line through $A$ perpendicular to $\overline{BC}$ intersects $\ell$ at $X$ and $BC$ at $Y$. The length of $\overline{XY}$ can be written in the form $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $100m+n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{962}
Calculate the cardinality of the set \(\left(\{1,2,3\}^{\mathbb{N}} - \{1,2\}^{\mathbb{N}}\right)\cap\mathcal{P}(\mathbb{N}\times\mathbb{N})\), where \(A^B\) denotes the set of all functions from \(B\) to \(A\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2^{\aleph_0}}
Calculate the total number of different possible staffing arrangements for three shot put match courts, where each court is staffed by two judges from different grades (10, 11, and 12), and there are two judges from each grade available. Provide your answer as a single whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{48}
Calculate the line integral of $\frac{1}{z}$ over a contour that consists of a square and a circle, both centered at the origin. Assume the contour is oriented counterclockwise.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\pi i}
In the expansion of \((-xy + 2x + 3y - 6)^6\), what is the coefficient of \(x^4 y^3\)? (Answer with a specific number)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["-21600"]}
Several irrational numbers are written on a blackboard. It is known that for every two numbers $ a$ and $ b$ on the blackboard, at least one of the numbers $ a\over b\plus{}1$ and $ b\over a\plus{}1$ is rational. What maximum number of irrational numbers can be on the blackboard? 

[i]Author: Alexander Golovanov[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Evaluate the contour integral \( \oint_{C} \left( z + \cfrac{z}{z+\cfrac{z}{\vdots}} \right) dz \), where \( C \) is a contour oriented counterclockwise around (but not including) the interval \([-4, 0]\) in the complex plane.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
Determine the number of distinct values of the expression

$$
\frac{n^{2}-2}{n^{2}-n+2}
$$

where \( n \in \{1, 2, \ldots, 100\} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{98}
For positive integers $n$, let $c_n$ be the smallest positive integer for which $n^{c_n} - 1$ is divisible by $210$, if such a positive integer exists, and $c_n = 0$ otherwise. What is $c_1 + c_2 + \dots + c_{210}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{329}
Let $C$ be the class of all real valued continuously differentiable functions $f$ on the interval $0<x<1$ with $f(0)=0$ and $f(1)=1$. Determine the largest real number $u$ such that
$$
u \leqslant \int_{0}^{1}\left|f^{\prime}(x)-f(x)\right| d x$$
for all $f$ in $C$. The original answer is in \(\frac{m}{ne}\) format, please provide the value of m+ n.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
在凸四边形 ABCD 中，\overrightarrow{BC}=2\overrightarrow{AD}，点 P 是四边形 ABCD 所在平面上一点，满足 \overrightarrow{PA}+2020\overrightarrow{PB}+\overrightarrow{PC}+2020\overrightarrow{PD}=\overrightarrow{0}. 设 s,t 分别为四边形 ABCD 与 \triangle PAB 的面积，则 \df{t}{s}=__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2358}
Find the interval $(n, n+1)$ (where $n$ is a natural number) within which the root of the function $f(x) = 3x - 7 + \ln(x)$ lies.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the minimum value of the expression $x^3 + y^3 + z^3$ subject to the constraints $x^2 + y^2 + z^2 = 1$ and $xy + xz + yz = 0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
a cricket bat is sold for $ 850 , making a profit of $ 215 . the profit percentage would be
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{33.86\%}
Given that $a > 0$, suppose proposition $p$: the function $y=a^{x}$ is monotonically increasing on $\mathbb{R}$; proposition $q$: the inequality $x^{2}-ax+1 > 0$ holds for all $x \in \mathbb{R}$. If $p$ and $q$ are false, and $\neg p$ is false, find the range of values for the real number $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[2,+\infty)}
Find minimum number \( n \) that:
1)\( {80} \mid  n \)
2) we can permute 2 different numbers in \( n \) to get \( m \) and \( {80} \mid  m \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
$n=800$
}
An ancient Greek was born on January 7, 40 B.C., and died on January 7, 40 A.D. How many years did he live?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{79}
Let \( z_{1}, z_{2}, \cdots, z_{12} \) be the roots of the equation \( z^{12}=2^{36} \). Let \( w_{j} (j=1,2, \cdots, 12) \) be one of \( z_{j} \) or \( iz_{j} \). Find the maximum value of \( \operatorname{Re} \sum_{j=1}^{12} w_{j} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16(1+\sqrt{3})}
Example 4-24 Use two colors to color the 6 faces and 8 vertices of a cube. How many different schemes are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{776}
Given $f(x)=\sin {x}^{2} - \sqrt{3} \cos x\cos \left(x+ \frac{π}{2}\right)$, find the interval(s) on which $f(x)$ is monotonically increasing within $\left[0, \frac{π}{2}\right]$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left[0, \frac{\pi}{3}\right]}
Two people take turns breaking a chocolate bar that is 6×8. During a move, they are allowed to make a straight-line break along any of the grooves. The player who cannot make a move loses. Who wins this game?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{First player}
Given a regular triangular pyramid $P-ABC$ (the base triangle is an equilateral triangle, and the vertex $P$ is the center of the base) with all vertices lying on the same sphere, and $PA$, $PB$, $PC$ are mutually perpendicular, and the side length of the base equilateral triangle is $\sqrt{2}$, calculate the volume of this sphere.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}\pi}{2}}
In the sequence $\{a_n\}$, $a_n = n^2 - kn$. If for any positive integer $n$, $a_n \geq a_3$ holds, then the range of values for $k$ is.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[5, 7]}
In a regular tetrahedron \( P-ABCD \) with lateral and base edge lengths both equal to 4, find the total length of all curve segments formed by a moving point on the surface at a distance of 3 from vertex \( P \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["6\\pi"]}
On a line \( l \) in space, points \( A \), \( B \), and \( C \) are sequentially located such that \( AB = 18 \) and \( BC = 14 \). Find the distance between lines \( l \) and \( m \) if the distances from points \( A \), \( B \), and \( C \) to line \( m \) are 12, 15, and 20, respectively.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Let $S=\left\{ \frac{p}{q} \mid q \leq 2009, \frac{p}{q} < \frac{1257}{2009}, p,q \in \mathbb{N} \right\}$. If the maximum element of $S$ is $\frac{p_0}{q_0}$ in reduced form, find $p_0+q_0$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{595}
If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m \ne 0$, $n \ne 0$, then the sum of the roots is:
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Barney's grocery store sold out all of its items at the beginning of the pandemic, so they ordered extra items to restock the shelves. They sold another 1561 items that day, and have 575 items in the storeroom. They have 3,472 items left in the whole store. How many items did they order to restock the shelves?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5608}
Jack bought an ice cream cone before jogging to the beach. The ice cream cone will normally melt in 10 minutes. However, the weather conditions are affecting the melting rate of the ice cream: there's a 15mph wind blowing in the opposite direction of Jack's running path, and the temperature is 85°F, causing the ice cream to melt 25% faster than in normal conditions. The beach is 16 blocks away, and each block is 1/8th of a mile. How fast does Jack need to jog (in miles per hour) to get to the beach before the ice cream melts, considering the wind and temperature factors?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
In a school of wizardry, there are 13 students. Before a clairvoyance exam, the teacher seated them around a round table and asked each to guess who would receive the diploma of clairvoyance. Each student humbly did not mention themselves or their two neighbors and wrote: "None of these ten will receive it!" Of course, all those who passed the exam guessed correctly, and all the other students guessed incorrectly. How many wizards received the diploma?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Two circles with radii \(\sqrt{19}\) and \(\sqrt{76}\), which touch each other externally, are inscribed in a semicircle (i.e., each of the circles touches this semicircle and its diameter). Find the radius of the semicircle.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4\sqrt{19}}
A person walks on a 400-meter circular track at a speed of 6 kilometers per hour. They start from point A and walk clockwise for 1 minute, then counter-clockwise for 3 minutes, and again clockwise for 5 minutes. To return to the starting point A, the minimum time required is     minutes.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
In the isosceles trapezoid \( KLMN \), the base \( KN \) is equal to 9, and the base \( LM \) is equal to 5. Points \( P \) and \( Q \) lie on the diagonal \( LN \), with point \( P \) located between points \( L \) and \( Q \), and segments \( KP \) and \( MQ \) perpendicular to the diagonal \( LN \). Find the area of trapezoid \( KLMN \) if \( \frac{QN}{LP} = 5 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7\sqrt{21}}
In hexagon $ABCDEF$, $AC$ and $CE$ are two diagonals. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $\frac{AM}{AC}=\frac{CN}{CE}=r$. Given that points $B$, $M$, and $N$ are collinear, find $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Let $\mathbb{Z}_{\ge 0}$ denote the set of nonnegative integers.

Define a function $f:\mathbb{Z}_{\ge 0} \to \mathbb{Z}$ with $f\left(0\right)=1$ and 
\[ f\left(n\right)=512^{\left\lfloor n/10 \right\rfloor}f\left(\left\lfloor n/10 \right\rfloor\right) \]
for all $n \ge 1$. Determine the number of nonnegative integers $n$ such that the hexadecimal (base $16$) representation of $f\left(n\right)$ contains no more than $2500$ digits.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10010}
Determine the common rational root \( k \) of the following polynomial equations which is not integral:
\[45x^4 + ax^3 + bx^2 + cx + 8 = 0\]
\[8x^5 + dx^4 + ex^3 + fx^2 + gx + 45 = 0\]
This root \( k \) is assumed to be a negative non-integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\frac{1}{3}}
In the isosceles trapezoid ABCD, where AB is parallel to DC, AB = 998, DC = 1001, and AD = 1999, find the number of points P on segment AD such that ∠BPC = 90°. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
9. The real quartic $P x^{4}+U x^{3}+M x^{2}+A x+C$ has four different positive real roots. Find the square of the smallest real number $z$ for which the expression $M^{2}-2 U A+z P C$ is always positive, regardless of what the roots of the quartic are.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16}
How many numbers are in the list $-50, -44, -38, \ldots, 68, 74$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Find the smallest number $x$ such that $10^x \equiv 1 \pmod{1997}$, using Euler's totient function, which indicates $\phi(1997) = 1996 = 4 \times 499$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{998}
Two congruent right circular cones each with base radius $5$ and height $12$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. Determine the maximum possible value of the radius $r$ of a sphere that lies within both cones.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{40}{13}}
Find the mass percentage of Cl in Barium chloride
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{34.04\%}
A store sells three items: a radio, a television, and a refrigerator. The cost prices for these items are Rs. 1500, Rs. 8000, and Rs. 25000, respectively. The items were sold at the following prices: Rs. 1110, Rs. 7500, and Rs. 23000, respectively. Calculate the overall loss percentage for the store.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8.38\%}
Find the constant term in the expansion of \((1+x)(e^{-2x}-e^{x})^{9}\). Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{84}
In the regular tetrahedron \( P-ABC \), \( AB = 1 \), \( AP = 2 \). The plane \( \alpha \) passing through \( AB \) bisects its volume. Find the cosine of the angle between edge \( PC \) and plane \( \alpha \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{3\sqrt{5}}{10}}
$y''' + y'' -y' -y=0$ $y(0)=7,y'(0)=-3,y''(0)=\alpha$ Find all values of $\alpha$ for which the solution obeys $y(x) \rightarrow 0 $ as $x \rightarrow \infty$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-1}
Let $T:l_2 \to l_2$ be defined as $T((x_1,x_2,\ldots,x_n,\ldots))=(x_2-x_1,x_3-x_2,\ldots,x_{n+1}-x_n,\ldots)$. Determine the range of possible values for $\|T\|$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Let $a>0$ be a real number, and $\{x_n\}$ be a sequence such that $x_1=1$ and $ax_n=x_1+x_2+\cdots+x_{n+1}$ for all $n\ge 1$. Find the minimum value of $a$ such that $x_n>0$ for all $n\ge 1$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Timmy's parents have a 2 car garage, which has both cars inside it. Also inside is a riding lawnmower, a bicycle for Timmy as well as each of his parents, a tricycle for Timmy's little brother Joey, a unicycle that Timmy's dad practices riding on, a skateboard, a wheelbarrow, and a four-wheeled wagon for Timmy and Joey to play with. How many wheels in total are in this garage?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31}
已知 $[x]$ 表示不超过 $x$ 的整数, 如 $[1.2]=1,[-1.2]=-2$. 已知 $\alpha=\frac{1+\sqrt{5}}{2}$, 则 $\left[\alpha^{12}\right]=$ $\qquad$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{321}
A monkey climbs up and down a ladder with $n$ steps, ascending 18 steps at a time or descending 10 steps. If it can climb from the ground to the top step and then return to the ground, what is the minimum value of $n$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{26}
In a rectangle of size $3 \times 4$, 4 points are chosen. Find the smallest number $C$ such that the distance between some two of these points does not exceed $C$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2.5}
A circle with radius $R$ is drawn through the vertices $A$, $B$, and $C$ of a right trapezoid $ABCD$ (with $\angle A = \angle B = 90^\circ$). The circle intersects segments $AD$ and $CD$ at points $M$ and $N$ respectively, such that $AM : AD = CN : CD = 1 : 3$. Find the area of the trapezoid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4 \sqrt{5} R^2}{3}}
A circle is inscribed in a square, and within this circle, a smaller square is inscribed such that one of its sides coincides with a side of the larger square and two vertices lie on the circle. Calculate the percentage of the area of the larger square that is covered by the smaller square.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50\%}
If two lines \( l \) and \( m \) have equations \( y = -2x + 8 \), and \( y = -3x + 9 \), what is the probability that a point randomly selected in the 1st quadrant and below \( l \) will fall between \( l \) and \( m \)? Express your answer as a decimal to the nearest hundredth.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.16}
Five people are sitting at a round table. Let $f\geq 0$ be the number of people sitting next to at least 1 female and $m\geq0$ be the number of people sitting next to at least one male. Find the number of possible ordered pairs $(f,m)$. Express your answer as a single integer.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{8}
Given an $m$ by $n$ matrix $A$ whose elements are all 0 or 1, and each row/column of $A$ is weakly monotone, how many such matrices are there?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{{m+n\choose n}}