inclusionAI/Ring-lite-rl-data huggingface dataset:
Let $X_i$, $i \geq 0$ be independent and identically distributed random variables with probability mass function $p(j) = P\{X_i=j\}$ for $j=1,...,m$, where $\sum^{m}_{j=1}P(j)=1$. Find $E[N]$, where $N=\min\{n>0:X_n=X_0\}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{m}
Given f(x) = ax^2 + bx + c where a, b, c ∈ ℝ and a > 100, what is the maximum number of different integer values of x that satisfy |f(x)| ≤ 50?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Find the limit \( \lim\limits_{x \to \infty} \int_0^{x} \cos\left(\dfrac{\pi t^2}{2}\right) \, dt \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{1}{2}}
Determine the asymptotic behavior of the recurrence relation \( T(n) = T(n-1) + (n-1)! \cdot n^k \) in terms of \( \Theta \) or \( O \) notation, where \( k \) is usually 0, 1, or 2.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\Theta(n! \cdot n^{k-1})}
Calculate the limit: \[ \lim_{n\rightarrow \infty} \sqrt{n}\int_{0}^{\infty} \frac{1-\cos(x)}{1+nx^6}dx \]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{\pi}{12}}
Determine the maximum number of points that can be placed inside a semicircle of radius $r$ such that no two points are closer than $r$ apart. The points can be placed anywhere within the semicircle, including on the straight line, inside the area, or on the circumference.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Determine the number of critical points of the function \( h(x) = e^{3f(x) - 2x + 1} + e^{2f(x) + 3x - 2} \), given that \( f(x) \) has a minimum value of \(-3\) at \( x = 0 \) and a maximum value of \(3\) at \( x = -2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1}
Evaluate the integral:
$$\int_{0}^{\pi/2} \frac{\sin x \cos^5 x}{(1-2\sin^2x\cos^2x)^2}dx$$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\(\frac{\pi}{8}\)}
The diagonals of a quadrilateral $ABCD$ inscribed in a circle intersect at point $E$. Given that $\angle ADB = \frac{\pi}{8}$, $BD = 6$, and $AD \cdot CE = DC \cdot AE$, find the area of quadrilateral $ABCD$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9\sqrt{2}}
Consider a set of vectors $\{ x_i \}_{i=1}^{m}$ in $\mathbb{R}^d$ that span a subspace $S$ of dimension $n$. Let $\mathcal{S}$ be the set of matrices $A \in \mathbb{R}^{d\times d}$ that map each vector $x_i$ to itself. Determine the cardinality of $\mathcal{S}$ in terms of $n$ and $d$. Express your answer as a function of $n$ and $d$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{d(d - n)}
Given $0\leqslant a+b \lt 1$, $2\leqslant a-b \lt 3$, find the range of $b$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{(-\frac{3}{2}, -\frac{1}{2})}
长度分别为 $1, a, a, a, a, a$ 的线段能成为同一个四面体的 6 条棱的充要条件是 $a$ 的值满足什么条件？原始答案的形式为 a>\frac{\sqrt{k}}{m}，请计算k+m的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let \( x, y, z, w \in [0,1] \). Find the maximum value of \( S = x^2 y + y^2 z + z^2 w + w^2 x - xy^2 - yz^2 - zw^2 - wx^2 \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{8}{27}}
Evaluate the integral: $ \int^{+\infty}_{-\infty} \frac{x^2}{{(x^2-8x+20)}^2} \, dx $
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{$\frac{5}{4} \pi$}
Find all integers that are coprime with every term of the sequence defined by \( u_{n} = 2^{n} + 3^{n} + 6^{n} - 1 \) for \( n \in \mathbb{N}^{*} \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\{-1, 1\}}
If 3 ^ x * 4 ^ y / 2 ^ z = 59,049, x – y + 2z = 10, and x^2 + y^2 = z^2 , then x = ?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{10}
Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $\mathscr{S}$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Two subgroups $H$ and $K$ in $\mathscr{S}$ are called equivalent if they are conjugate in $G$. How many equivalence classes are there in $\mathscr{S}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Given that $\tan \alpha= \dfrac {1}{2}$, find the value of: $\dfrac {1+2\sin (\pi-\alpha)\cos (-2\pi-\alpha)}{\sin ^{2}(-\alpha)-\sin ^{2}( \dfrac {5\pi}{2}-\alpha)}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-3}
Evaluate the limit: $$\lim_{x\to 0} \frac{e^x-\cos x-\ln(1+x)}{x^4}$$ using the squeeze theorem or L'Hopital's rule.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\infty}
Joseph has a refrigerator, a water heater, an electric oven, an air conditioner, and a washing machine in his house that consumes power at different rates. The total amount of money that Joseph pays for the energy used by the refrigerator is three times the amount he pays for the power used by the water heater. The electric oven uses power worth $500 in a month, 2.5 times what Joseph pays for the power the water heater uses. The air conditioner consumes power worth $300 in a month, which is 75% the cost of the power used by the refrigerator. The washing machine consumes power worth $100 in a month, which is one-quarter of the cost of the power used by the water heater. Calculate the total amount Joseph pays for the power used by all these gadgets in a month, while considering the aforementioned ratios and relationships between the appliances.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\$1700}
If the graph of the power function $f(x)=x^{\alpha}$ ($\alpha$ is a constant) always passes through point $A$, and the line ${kx}{-}y{+}2k{+}1{+}\sqrt{3}{=}0$ always passes through point $B$, then the angle of inclination of line $AB$ is _____.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{5\pi}{6}}
Determine the number of terms required in the series \( e = \sum_{k=0}^{\infty} \frac{1}{k!} \) to approximate \( e \) with an error of at most \( \frac{6}{10} \) unit in the 20th decimal place.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{22}
Given the circle $C: x^{2}+(y+5)^{2}=3$, how many lines are there that are tangent to the circle and have equal y-intercepts and x-intercepts? Express your answer as a whole number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A set $A$ of positive integers is called [i]uniform[/i] if, after any of its elements removed, the remaining ones can be partitioned into two subsets with equal sum of their elements. Find the least positive integer $n>1$ such that there exist a uniform set $A$ with $n$ elements.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{7}
Compute the number of five-digit positive integers whose digits have exactly $30$ distinct permutations (the permutations do not necessarily have to be valid five-digit integers).

[i]Proposed by David Sun[/i]
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{9720}
Evaluate the triple integral \( \iiint \frac{1}{\sqrt{x^2+y^2+(z-2)^2}} \, dV \) over the region defined by \( x^2+y^2+z^2 = 1 \) using spherical coordinates.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\dfrac{2\pi}{3}}
Let $S=\{1,2, \ldots, 9\}$. Compute the number of functions $f: S \rightarrow S$ such that, for all $s \in S, f(f(f(s)))=s$ and $f(s)-s$ is not divisible by 3.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{288}
In how many ways can 7 distinct balls be packed into 16 boxes, where 7 of the boxes are unlabeled and green, and 9 of the boxes are unlabeled and yellow?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{14214}
Define $f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}$. Find the smallest positive integer $x$ such that $f(x) \ge 50\sqrt{x}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2400}
A circular disk is divided by $2n$ equally spaced radii ($n>0$) and one secant line. The maximum number of non-overlapping areas into which the disk can be divided is
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{3n+1}
[Dirichlet's Principle (finite number of points, lines, etc.)]

In a park, there are 10,000 trees planted in a square grid (100 rows of 100 trees each). What is the maximum number of trees that can be cut down so that the following condition is met: if you stand on any stump, you will not see any other stump? (The trees can be considered sufficiently thin.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2500}
Given point \( A(3,1) \) and point \( F \) as the focus of the parabola \( y^{2}=5x \), let \( M \) be a variable point on the parabola. When \( |MA| + |MF| \) takes its minimum value, the coordinates of point \( M \) are _______.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\left(\frac{1}{5}, 1\right)}
If $P$ and $Q$ are invertible matrices such that $PQ = -QP$, and the trace of $P$ is $-2$, determine the trace of $Q$. Provide your answer as a single number.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
已知函数 $f(x)=-x^{3}+a x^{2}+b x+c(a, b, c \in \mathbf{R})$ 在 $(-\infty, 0)$ 上单调递减, 在 $(0,1)$ 上单调递增,且 $f(x)$ 在 $\mathbf{R}$ 上有三个零点, 1 是其中的一个零点.求 $f(2)$ 最小值的 $2 $倍是多少
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-5}
There are 11 of the number 1, 22 of the number 2, 33 of the number 3, and 44 of the number 4 on the blackboard. The following operation is performed: each time, three different numbers are erased, and the fourth number, which is not erased, is written 2 extra times. For example, if 1 of 1, 1 of 2, and 1 of 3 are erased, then 2 more of 4 are written. After several operations, there are only 3 numbers left on the blackboard, and no further operations can be performed. What is the product of the last three remaining numbers?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
If $10x+y=75$ and $10y+x=57$ for some positive integers $x$ and $y$, what is the value of $x+y$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Consider sequences of positive real numbers of the form $x, 2000, y, \ldots,$ in which every term after the first is 1 less than the product of its two immediate neighbors. Find the number of different values of $x$ for which the term 2001 appears somewhere in the sequence.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
Two particles are positioned at opposite vertices of the same face of a cube. If at a given time \( t \) the particles are on the same edge, they remain in the same position up to time \( t + 1 \). Otherwise, one of them is chosen at random with equal probability, and at time \( t + 1 \) it is moved to one of the three neighboring vertices with equal probability. What is the expected value of the least time they are on the same edge?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
Jack is reading a 285-page book. On weekdays, he reads 23 pages per day, but on weekends, he reads 35 pages per day. How many days will it take him to finish reading the book?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{12}
Find the smallest positive integer \( n \) such that:
1. \( n \) has exactly 144 distinct positive divisors.
2. There are ten consecutive integers among the positive divisors of \( n \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{110880}
In a simplified version of the game of werewolf, there are two werewolves and two townsfolk. The werewolves know everyone's identities, but the townsfolk only know their own. A voting order is randomly determined, and each player votes against one of the other three players. The townsfolk win if at least one of the players with the most votes is a werewolf; otherwise, the werewolves win. Assuming optimal play from both teams, what is the probability for the townspeople to win?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2/3}
共有 11 个剧团参加汇演, 每天都排定其中某些剧团演出, 其余的剧团则跻身于普通观众之列。在汇演结束时, 每个剧团除了自己的演出日外,至少要看过其它每个剧团的一次表演。问: 这样的汇演至少要安排几天?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{6}
Let $A$ be a $4 \times 4$ invertible matrix with real coefficients. If $B = A^T A^{-1}$ and $\det(B) = 2$, find $\det(I - B)$. \( \boxed{?} \)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{0}
CJ, KJ, AJ, and BJ collect stamps. CJ has 5 more than twice the number of stamps that KJ has, KJ has half as many as AJ, and BJ has 3 less than thrice the number of stamps AJ has. If the four of them have 1,472 stamps all together, how many stamps does AJ have?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{267}
Anders is solving a math problem and encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$, where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q \cdot 15!$ for some rational number $q$. Find $q$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
The base of a quadrilateral pyramid is a square \(ABCD\) with each side equal to 2. The lateral edge \(SA\) is perpendicular to the base plane and also equals 2. A plane is passed through the lateral edge \(SC\) and a point on side \(AB\) such that the resulting cross-section of the pyramid has the smallest perimeter. Find the area of this cross-section.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\sqrt{6}}
If the height of a cone is increased by 140% and its radius is decreased by 50%, by what percentage does its volume change?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{40\%}
Bogdanov I.I.
A chess piece can move 8 or 9 squares horizontally or vertically. It is forbidden to move to the same square twice.
What is the maximum number of squares this piece can visit on a $15 \times 15$ board? (The tour can start from any square.)
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{196}
Find the number of subsets $A \subset M = \{2^0, 2^1, 2^2, \dots, 2^{2005}\}$ such that the equation $x^2 - S(A)x + S(B) = 0$ has integral roots, where $S(M)$ is the sum of all elements of $M$, and $B = M \setminus A$ ($A$ and $B$ are not empty).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1003}
A point $Q$ is randomly chosen inside an equilateral triangle $DEF$. What is the probability that the area of $\triangle DEQ$ is greater than both $\triangle DFQ$ and $\triangle EFQ$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{1}{3}}
In the context of the Riemann integral, consider a function $f(x)$ defined over the interval $[0,1]$. What is the cardinality of the set of points where $f(x)$ is evaluated when computing the integral $\int_{0}^{1} f(x) \, dx$? Assume the continuum hypothesis holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\aleph_1}
What is the minimum number of cubes needed to write one digit on each face such that any ordered combination of three digits from 000 to 999 can be obtained by selecting three different cubes and placing them with suitable faces up in the correct order?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{5}
Find the completion of the metric space \((\ell^1(\mathbb{N}), d_*)\), where \(\ell^1(\mathbb{N}) = \{(a_n): a_n \in \mathbb{R}, \sum |a_n| < \infty \}\) and the distance function \(d_*\) is defined by \(d_*(a,b) = \sup |a_n - b_n|\) for \(a, b \in \ell^1(\mathbb{N})\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{c_0}
In a regular hexagon $A B C D E F$, the diagonals $A C$ and $C E$ are divided by points $M$ and $N$ respectively in the following ratios: $\frac{A M}{A C} = \frac{C N}{C E} = r$. If points $B$, $M$, and $N$ are collinear, determine the ratio $r$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{\sqrt{3}}{3}}
Find all functions \( f: \mathbb{R} \to \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the equation \( f(xy) = f(f(x) + f(y)) \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{f(x) = c}
令 $a_{n}$ 表示距离 $\sqrt{n}$ 最近的正整数，其中 $n \in \mathbf{N}^{*}$ 。若 $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=2016$，则求正整数 $n$ 的值。
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1017072}
Moe has a new, larger rectangular lawn measuring 120 feet by 180 feet. He uses a mower with a swath width of 30 inches. However, he overlaps each cut by 6 inches to ensure no grass is missed. Moe walks at a rate of 6000 feet per hour while pushing the mower. What is the closest estimate of the number of hours it will take Moe to mow the lawn?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1.8}
In a rectangular parallelepiped \(ABCDEFGHA'B'C'D'\), which has \(AB = BC = a\) and \(AA' = b\), the shape was orthogonally projected onto some plane containing the edge \(CD\). Find the maximum area of the projection.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{a \sqrt{a^2 + b^2}}
Two weightless glasses with different quantities of water are balanced on the beam of unequal-arm scales with length $L$. An amount of water with mass $m$ is poured from one glass into the other, causing the scales to lose balance. To restore balance, the support point of the beam needs to be shifted by a distance $l$ relative to its original position. Find the total mass of water in the glasses, ignoring the mass of the beam itself.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{m L}{l}}
Given that in $\triangle ABC$, the internal angles $A$ and $B$ satisfy $\frac{\sin B}{\sin A} = 2\cos(A+B)$, find the size of angle $C$ when $B$ is at its maximum value.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{2\pi}{3}}
The following question appeared in my examination. Let $G$ be the symmetric group $S_5$ of permutations of five symbols. Consider the set $\mathscr{S}$ of subgroups of $G$ that are isomorphic to the non-cyclic group of order $4$. Let us call these two subgroups $H$ and $K$ belonging to $\mathscr{S}$ as equivalent if they are conjugate (that is, there exists $g \in G$ such that $gHg^{-1}=K$). How many equivalence classes are there in $\mathscr{S}$?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
The equation $ ax^3+bx^2+cx+d=0$ has three distinct solutions. How many distinct solutions does the following equation have:

$ 4(ax^3+bx^2+cx+d)(3ax+b)=(3ax^2+2bx+c)^2?$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{2}
A coin-making machine produces pennies, each with a probability $P$ of turning up heads, where $P$ is drawn from a uniform distribution on $[0,1]$. You flip a coin from this machine once, and it comes up heads. Determine the conditional distribution function $F_{P|H}(p)$ for the probability of a head for that coin, given this outcome.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p^2}
Determine the values of $p$ for which the curve $y = x^p \cos \frac{\pi}{x}$, defined for $0 < x \leq 1$, has finite length.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{p > 1}
Determine the number of homomorphisms from the group \( \mathbb{Z} \times (\mathbb{Z} / 6\mathbb{Z}) \) to the group \( (\mathbb{Z} /2\mathbb{Z} ) \times (\mathbb{Z} /60\mathbb{Z}) \).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1440}
On an island, there are knights, liars, and followers; each person knows who is who. All 2018 island residents were lined up and each was asked to answer "Yes" or "No" to the question: "Are there more knights than liars on the island?" The residents responded one by one in such a way that the others could hear. Knights always told the truth, liars always lied. Each follower answered the same as the majority of the preceding respondents, and if the "Yes" and "No" answers were split equally, they could give either answer. It turned out that there were exactly 1009 "Yes" answers. What is the maximum number of followers that could be among the island residents?
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1009}
Compare the sizes: $-(+\frac{5}{9}) \_\_\_\_\_\_-|-\frac{4}{7}|$
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{>}
Calculate the integral \( \int_0^1{\frac{\left(2x^3-3x^2\right)f'(x)}{f(x)}}\,dx \) given that the function \( f(x) \) is differentiable on \([0, 1]\), satisfies \( f(1) = 1 \), and the functional equation \( f(x)f(1-x) = e^{x^2 - x} \) holds.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{-\dfrac{1}{10}}
The probability that a set of three distinct vertices chosen at random from among the vertices of a regular $n$-gon determine an obtuse triangle is $\tfrac{93}{125}$. Find the sum of all possible values of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{503}
Evaluate the limit $\lim_{n \to \infty} nt_n$, where $t_n = \frac{1}{2n+1} - \frac{1}{2n+2} + \frac{1}{2n+3} - \frac{1}{2n+4} + \cdots + \frac{1}{4n-1} - \frac{1}{4n}$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{1/8}
The duration $Y$ of long-distance telephone calls (in minutes) is a random variable with the following properties: $P(Y=3)=0.2$ and $P(Y=6)=0.1$. Otherwise, $Y$ has a continuous density function given by \( f(y)= \begin{cases} (1/4)ye^{-y/2}, & y>0 \\ 0, & \text{elsewhere.} \end{cases} \) Find the expected duration of a randomly selected long-distance call.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{4}
A plane, perpendicular to an edge of a regular tetrahedron, passes through a point that divides this edge in the ratio 1:4. Find the ratio of the volumes of the resulting parts of the tetrahedron.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{\frac{4}{121}}
Find the smallest integer \(n > 1\) such that \(2^n > n^{1000}\).
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{13747}
A spider is making a web between $n>1$ distinct leaves which are equally spaced around a circle. He chooses a leaf to start at, and to make the base layer he travels to each leaf one at a time, making a straight line of silk between each consecutive pair of leaves, such that no two of the lines of silk cross each other and he visits every leaf exactly once. In how many ways can the spider make the base layer of the web? Express your answer in terms of $n$.
Please reason step by step, and put your final answer within \boxed{}.
Answer: \boxed{n 2^{n-2}}