inclusionAI/Ring-lite-rl-data huggingface dataset:
In a house, a hall is 20 m long, some meters wide, and 5 m high. Its interior has to be covered with mat. The total expenditure for covering the hall with mat at a cost of Rs. 20 per square meter is Rs. 19000. What is the width of the hall?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{25}
Determine the value of \( n \) such that the series \( 1\cdot(0)!+3\cdot(1!)+7\cdot(2!)+13\cdot(3!)+21\cdot(4!)+\cdots \) up to \( n \) terms equals \( (4000)4000! \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4000}
Given a Markov chain with state space $E = \{1,2,3,4,5\}$ and the transition matrix:

$$ P = \begin{bmatrix} 0 & \frac{1}{2} & \frac{1}{2} & 0 & 0 \\ \frac{1}{2} & 0 & \frac{1}{2} & 0 & 0 \\ \frac{1}{4} & \frac{1}{4} & 0 & \frac{1}{4} & \frac{1}{4} \\ 0 & 0 & \frac{1}{2} & 0 & \frac{1}{2} \\ 0 & 0 & 0 & \frac{1}{2} & \frac{1}{2} \end{bmatrix} $$

Assuming $X_1 = 1$, calculate the expected number of transitions between successive visits to state 1, given that state 5 is never visited.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Compute the second homotopy group \( \pi_2(X) \) for the space \( X = D^2 \cup_f S^1 \), where \( f : S^1 \to S^1 \) is a degree \( m \) map.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\mathbb{Z}}
Consider a random variable Y that is a mixture of a standard uniform random variable X and a degenerate random variable that always takes the value 1. Let p be a positive real number. Find the variance of Y^p as p approaches infinity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/4}
Riley has 64 cubes with dimensions \(1 \times 1 \times 1\). Each cube has its six faces labeled with a 2 on two opposite faces and a 1 on each of its other four faces. The 64 cubes are arranged to build a \(4 \times 4 \times 4\) cube. Riley determines the total of the numbers on the outside of the \(4 \times 4 \times 4\) cube. How many different possibilities are there for this total?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Let  $S$  be the set of triples  $(a,b,c)$  of non-negative integers with  $a+b+c$  even. The value of the sum
    \[\sum_{(a,b,c)\in S}\frac{1}{2^a3^b5^c}\]
    can be expressed as  $\frac{m}{n}$  for relative prime positive integers  $m$  and  $n$ . Compute  $m+n$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{37}
Evaluate the double integral \( I = \iint_{[0,1]^2} \frac{dx \, dy}{(1+x^2+y^2)^{3/2}}. \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{6}}
A pyramid has a base in the shape of a parallelogram with adjacent sides measuring 9 cm and 10 cm, and one of the diagonals is 11 cm. The opposite lateral edges of the pyramid are equal, and each of the longer edges measures 10.5 cm. Calculate the volume of the pyramid.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{200}
已知 $0<k<4$, 直线 $l_{1}: k x-2 y-2 k+8=0$ 和直线 $l_{2}: 2 x+k^{2} y-4 k^{2}-4=0$ 与两坐标轴围成一个四边形。求使这个四边形面积最小的 $k$ 值。原始答案的形式为$\frac{m}{n}$，请给出$m+n$的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9}
Find an upper bound for the number of solutions \( N_{\sqrt{2}}(1) \) to the inequality \( \left \lvert \sqrt{2} - \frac{p}{q} \right \rvert < \frac{1}{q^{3}} \) for \( \sqrt{2} \) with \( \epsilon=1 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
In a 100 x 100 grid, the numbers 1 to 10000 are placed such that each number appears exactly once, and numbers differing by 1 are in adjacent cells. Determine the maximum possible value of the minimum distance S between the centers of any two cells containing numbers that differ by exactly 5000.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{50}
Let $g(n)$ be defined as:

$$g(n)=\frac{\sum^{n}_{k=0}\cos\left(\frac{k+1}{n+2}\pi\right)\cos\left(\frac{k+2}{n+2}\pi\right)}{\sum^{n}_{k=0}\cos^2\left(\frac{k+1}{n+2}\pi\right)}$$

Evaluate $\lim_{n\to\infty} g(n)$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
If ( 1 / 3 ) ^ 32 ( 1 / 125 ) ^ k = 1 / 27 ^ 32, then k =
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
One side of a rectangular prism is 15 m, one of its diagonals is 17 m, and its height is h m. Find the volume of the prism in cubic meters.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{120h}
There are four people in a room. For every two people, there is a $50 \%$ chance that they are friends. Two people are connected if they are friends, or a third person is friends with both of them, or they have different friends who are friends of each other. What is the probability that every pair of people in this room is connected?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{19}{32}}
Determine the Lebesgue measure of a generalized Cantor set formed by removing the middle fifths from the unit interval.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
A person travels equal distances with varying speeds and conditions. They start by walking at a speed of 2 km/h on a flat terrain, then switch to riding a bicycle at a speed of 6 km/h uphill with a 5% incline, and finally skateboard at a speed of 6 km/h downhill with a 10% decline and facing a 2 km/h headwind. The person takes a total time of 11 minutes to travel the entire route. Determine the total distance traveled?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.6}
7. [6] Rachel has two indistinguishable tokens, and places them on the first and second square of a $1 \times 6$ grid of squares, She can move the pieces in two ways:
- If a token has free square in front of it, then she can move this token one square to the right
- If the square immediately to the right of a token is occupied by the other token, then she can "leapfrog" the first token; she moves the first token two squares to the right, over the other token, so that it is on the square immediately to the right of the other token.
If a token reaches the 6th square, then it cannot move forward any more, and Rachel must move the other one until it reaches the 5 th square. How many different sequences of moves for the tokens can Rachel make so that the two tokens end up on the 5 th square and the 6 th square?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{42}
Within a cube structure consisting of 8 vertices, the midpoints of the 12 edges, the centers of the 6 faces, and the center of the cube (totaling 27 points), determine the total number of groups of three collinear points.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{49}
Let  $x_0=1$ , and let  $\delta$  be some constant satisfying  $0<\delta<1$ . Iteratively, for  $n=0,1,2,\dots$ , a point  $x_{n+1}$  is chosen uniformly form the interval  $[0,x_n]$ . Let  $Z$  be the smallest value of  $n$  for which  $x_n<\delta$ . Find the expected value of  $Z$ , as a function of  $\delta$ .
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1 - \ln \delta}
Evaluate the integral \(\int_{\mathcal{C}(0,3)} \frac{e^{2/z}}{z-2}\,dz\) where \(\mathcal{C}(0,3)\) is the circle centered at the origin with radius 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i (1 + e)}
Evaluate the integral: $$\int_0^1\int_0^\infty ye^{-xy}\sin x\,dx\,dy$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\ln 2}{2}}
Given the series $\frac{1}{\sin 8^\circ} + \frac{1}{\sin 16^\circ} + \frac{1}{\sin 32^\circ} + \ldots + \frac{1}{\sin 4096^\circ} + \frac{1}{\sin 8192^\circ} = \frac{1}{\sin \alpha}$, where $\alpha \in (0, 90^\circ)$, find the value of $\alpha$ in degrees.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Determine the number of solutions to the equation \( 6 \ln(x^2+1) = e^x \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
Given 7 points \( A_{1}, A_{2}, \cdots, A_{7} \) on a plane, where no three points are collinear and their convex hull is the triangle \( \triangle A_{1} A_{2} A_{3} \), determine the number of convex quadrilaterals that can be formed by selecting 4 of these points as vertices.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Consider the set \( S = \{1, 2, 3, \ldots, 2022\} \) and a permutation \( A = \{a_{1}, a_{2}, a_{3}, \ldots, a_{2022}\} \) of \( S \). It is known that for any \(1 \leq n, m \leq 2022\), the sum \( a_{n} + a_{m} \) is divisible by the greatest common divisor (GCD) of \( n \) and \( m \). Determine the number of possible permutations \( A \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the interval of convergence for the Taylor series representation of the function \( f(x) = x^{0.5} \) centered at 6, given by \( \sum_{n=0}^\infty c_n(x-6)^n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{[0, 12]}
In an equilateral triangular pyramid with base \( \triangle ABC \), point \( P \) lies inside the triangle such that \( PA^2 = PB^2 + PC^2 \). Find the measure of \( \angle BPC \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{150^\circ}
Evaluate the integral \( I = \int_0^\infty \log\left(x + \frac{1}{x}\right) \frac{1}{1+x^2} \, dx \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\pi \ln 2}
Evaluate the triple integral \( \int \int \int (x-y) \, dV \) over the solid defined by the limits: \( y \in [0,2] \), \( z \in [-x^2+1, x^2-1] \), and bounded by the parabolic cylinders \( z = x^2 - 1 \) and \( z = 1 - x^2 \) in the xy-plane, forming a square with sides parallel to the coordinate axes and centered at \((0, 1)\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{16}{3}}
Evaluate the integral \( \int_{C_1} \frac{dz}{z} \) where \( C_1 \) is a contour from \( z=0 \) to \( z=R e^{2\pi i} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2\pi i}
The budget of the Petrov family consists of the following sources of income:

- Parents' salary after income tax deduction: 56000 rubles;
- Grandmother's pension: 14300 rubles;
- Son's scholarship: 2500 rubles.

The average monthly expenses of the family include:

- Payment for communal services: 9800 rubles;
- Food expenses: 21000 rubles;
- Transport expenses: 3200 rubles;
- Leisure: 5200 rubles;
- Other expenses: 15000 rubles.

10 percent of the remaining amount is transferred to a deposit for forming a financial safety cushion. Determine the amount that the Petrov family can save monthly for an upcoming vacation.

In the answer, indicate only the number without units of measurement!
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{16740}
Determine the number of distinct permutations of the letters in the word 'LYCANTHROPIES' such that the letter 'C' is not adjacent to 'A', 'A' is not adjacent to 'N', and 'N' is not adjacent to 'T'.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3665088000}
Given $e_{60}=17$, $_{10}P_{50}=0.8$, and the $10$-year curtate temporary life expectancy at age $50$ is $9.2$, find $e_{50}$ using the recursive formula $e_x = P_x(1 + e_{x+1})$. Calculate $e_{50}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22.8}
Real numbers \( x_{1}, x_{2}, \cdots, x_{2001} \) satisfy \( \sum_{k=1}^{2000} \left| x_{k} - x_{k+1} \right| = 2001 \). Let \( y_{k} = \frac{1}{k} \left( x_{1} + x_{2} + \cdots + x_{k} \right) \) for \( k = 1, 2, \cdots, 2001 \). Find the maximum possible value of \( \sum_{k=1}^{2000} | y_{k} - y_{k+1} | \). (2001 Shanghai Mathematics Competition)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2000}
Evaluate the line integral \( \oint _{ C }^{ }{ \left( { e }^{ x }\sin { y } \right) dx+\left( { e }^{ x }\cos { y+x } \right) dy } \), where \(C\) is the arc of the circumference \( x^2 + y^2 = 1 \) in the first quadrant, traversed in the counterclockwise direction.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\pi}{4} + \sin 1}
Two swimmers start at opposite sides of a 90-foot swimming pool. One swims at 3 feet per second, while the other swims at 2 feet per second. They swim back and forth for 12 minutes without taking any time to turn around. How many times do they meet? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{20}
Determine the largest natural number $m$ such that for each non-negative real numbers $a_1 \ge a_2 \ge \ldots \ge a_{2014} \ge 0$, the following inequality holds: $$\frac{a_1 + a_2 + \ldots + a_m}{m} \ge \sqrt{\frac{a_1^2 + a_2^2 + \ldots + a_{2014}^2}{2014}}.$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{44}
Let x be the number of crayons Denise gives to each of her 30 friends, such that the ratio x/210 represents the proportion of crayons each friend receives. If Denise gives away 70% of her crayons among her 30 friends, what is the value of x that corresponds to the number of crayons each friend gets?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4.9}
已知函数$f\left( x \right)=\frac{x{{e}^{x}}+x+2}{{{ex}}+1}+\sin x$，则$f\left( -5 \right)+f\left( -4 \right)+f\left( -3 \right)+f\left( -2 \right)+f\left( -1 \right)+f\left( 0 \right)+f\left( 1 \right)+f\left( 2 \right)+f\left( 3 \right)+f\left( 4 \right)+f\left( 5 \right)$的值是________.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{11}
Given the function f(x) = ax^3 - 3x^2 + 1, if f(-a), f(a), and f(3a) form an arithmetic sequence with a non-zero common difference, calculate the number of tangents that can be drawn from the origin to the curve y = f(x).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
I want to choose a license plate which is 3 characters long, where the first character is a letter, the last character is a digit, and the middle is either a letter or a digit. I also want there to be two characters on my license plate which are the same. How many ways are there for me to choose a license plate with these restrictions?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{520}
In a basketball game, a certain team played a total of 8 games and scored 25, 29, 30, 32, 37, 38, 40, 42 points respectively. What is the 65th percentile of this data set?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{38}
Evaluate the limit of the integral as \( R \) approaches infinity: \[ \lim_{R \to \infty} \frac{1}{2\pi i}\int_{-R}^{R}\frac{1}{t-(a-bi)}dt \] where \( a \) and \( b \) are real constants and \( b > 0 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{2}}
$-{{8}^{\frac{2}{3}}}+{{\log }_{3}}\sqrt{27}+2\lg 5+\lg 4+{{7{{\log 7}}2}}=$______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{3}{2}$}
Randomly divide the positive integers $1, 2, 3, 4, 5, 6, 7$ into two groups, ensuring that each group contains at least one number. Calculate the probability that the sum of the numbers in each group is equal. Express your answer as a fraction in simplest form.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{4}{63}}
Find the area of an isosceles trapezoid, given that its height is \( h \) and its lateral side is visible from the center of the circumscribed circle at an angle of \( 60^{\circ} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{h^{2} \sqrt{3}}
Revise the expression to accurately bound the probability $P(X > t\alpha)$, where $X$ follows a Poisson distribution with parameter $\alpha = N/M$, using Markov's inequality.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{t}}
From a point A, the bearing to the base of a 60m high tower is 53°, and the angle of elevation to the top of the tower is 16°. From a point B, the bearing to the base of the tower is 300°, and the angle of elevation to the top is 20°. Calculate the distance between points A and B.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{313}
Determine the number of distinct cyclic subgroups of order 12 in the group \( \mathbb{Z}_6 \times \mathbb{Z}_{10}^\times \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
In an equilateral triangle \(ABC\) with side length \(a\), points \(M, N, P, Q\) are located such that \(MA + AN = PC + CQ = a\). Determine the measure of angle \(NOQ\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{60^\circ}
Calculate the integral \( \int_{0}^{1} f(x) \, dx \) for a function \( f \), where \( f \) is continuous on \([0,1]\), \( f(1) = 0 \), \( \int_{0}^{1} [f'(x)]^2 \, dx = 7 \), and \( \int_{0}^{1} x^2 f(x) \, dx = \frac{1}{3} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{7}{5}}
Let $x_1, x_2, ... , x_n$ be a real numbers such that
1) $1 \le x_1, x_2, ... , x_n \le 160$
2) $x^{2}_{i} + x^{2}_{j} + x^{2}_{k} \ge 2(x_ix_j + x_jx_k + x_kx_i)$ for all $1\le i < j < k \le n$
Find the largest possible $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{
n = 6
}
Given a function $H(w)$, determine an upper bound on the Lipschitz constant of this function, assuming that $\|x_i\| \leq 1$ for all $i$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Determine the value of $x$ at which the Bessel equation $x^2y'' + xy' + (x^2 - n^2)y = 0$ has a regular singularity.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
Determine the largest possible value of n such that for any permutation of n points in the plane, the resulting broken line does not intersect itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3}
A random glass breaks at an exponentially distributed time, with a mean of 24 months. Calculate the probability that, out of 500 glasses, at most 210 break within a year.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0.896}
Compare the numbers: \(9^{8^{8^9}}\) and \(8^{9^{9^8}}\). Determine which is larger.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9^{8^{8^9}}}
an empty fuel tank is filled with brand z gasoline . when the tank is 3 / 4 empty , it is filled with brand y gasoline . when the tank is half empty again , it is filled with brand z gasoline . when the tank is half empty again , it is filled with brand y gasoline . at this time , what percent of the gasoline in the tank is brand z ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{31.25\%}
On the radius $AO$ of a circle with center $O$, a point $M$ is chosen. On one side of $AO$, points $B$ and $C$ are chosen on the circle such that $\angle AMB = \angle OMC = \alpha$. Find the length of $BC$ if the radius of the circle is 21 and $\sin \alpha = \frac{\sqrt{40}}{7}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{18}
Some men can check exam papers in 8 days working 5 hours regularly. The total hours when 2 men will check the double of the papers in 20 days is 8. How many men were initially checking the exam papers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The center of the circle inscribed in a trapezoid is at distances of 5 and 12 from the ends of one of the non-parallel sides. Find the length of this side.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13}
A straight line is drawn across an 8 by 8 checkerboard. What is the greatest number of 1 by 1 squares through which this line could pass?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{15}
On a rotating round table, there are 8 white teacups and 7 black teacups. Fifteen dwarves wearing hats (8 white hats and 7 black hats) are sitting around the table. Each dwarf picks a teacup of the same color as their hat and places it in front of them. After this, the table is rotated randomly. What is the maximum number of teacups that can be guaranteed to match the color of the dwarf's hat after the table is rotated? (The dwarves are allowed to choose their seating, but they do not know how the table will be rotated.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Farm tax is levied on the 45% of the cultivated land. The tax department collected a certain amount through the farm tax from the village of Mr. Willam. Mr. Willam paid only $480 as farm tax. The percentage of total land of Mr. Willam over the total taxable land of the village is 27.77777777777778%. How much money did the tax department collect from the village?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$1728}
The brothers found a treasure of gold and silver. They divided it so that each got 100 kg. The eldest got the most gold - 25 kg - and one-eighth of all the silver. How much gold was in the treasure?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{100}
A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be critical if for each $i,j \in S$ at least one of the four points $(i,j), (i + 1,j), (i,j + 1)$, and $(i + 1, j + 1)$ $(99 + 1 \equiv 0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5000}
Given $a \gt 0$, if the inequality $8{e}^{ax}(ln2+\frac{ax}{2})-xlnx⩾0$ holds for any $x∈[\frac{1}{e},+∞)$, then the minimum value of $a$ is ______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{4e}}
Given two lines $l_{1}: ax-y+a=0$ and $l_{2}: (2a-3)x+ay-a=0$ are parallel, determine the value of $a$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Given that $5^{2018}$ has 1411 digits and starts with 3 (the leftmost non-zero digit is 3), for how many integers $1 \leq n \leq 2017$ does $5^n$ start with 1?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{607}
What will be the length of the strip if a cubic kilometer is cut into cubic meters and laid out in a single line?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1000000}
8. Given the set of lattice points $S=\{(x, y) \mid x, y \in\{0,1,2,3,4,5\}\}$, find the number of triangles with vertices in $S$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{["6768"]}
The product of the digits of a 5-digit number is 180. How many such numbers exist?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{360}
Calculate the variance of the number of derangements in a random permutation of $n$ elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}