Let $a_n=\binom{200}{n}(\sqrt[3]{6})^{200-n}\left(\frac{1}{\sqrt{2}}\right)^n$, where $n=1$, $2$, $\cdots$, $95$. Find the number of integer terms in $\{a_n\}$.
Randomly choosing two numbers from the set $\{1, 3, 5, 7, 9\}$ with replacement, what is the probability that the product is greater than 40?
A book contains $250$ pages. How many times is the digit used in numbering the pages?
Three darts are thrown at at $3\times 3$ target, each landing in a different square. What is the probability that the squares they land in form a row, either horizontally, vertically or diagonally?
Two dice appear to be normal dice with their faces numbered from $1$ to $6$, but each die is weighted so that the probability of rolling the number $k$ is directly proportional to $k$. The probability of rolling a $7$ with this pair of dice is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
A regular icosahedron is a $20$-faced solid where each face is an equilateral triangle and five triangles meet at every vertex. The regular icosahedron shown below has one vertex at the top, one vertex at the bottom, an upper pentagon of five vertices all adjacent to the top vertex and all in the same horizontal plane, and a lower pentagon of five vertices all adjacent to the bottom vertex and all in another horizontal plane. Find the number of paths from the top vertex to the bottom vertex such that each part of a path goes downward or horizontally along an edge of the icosahedron, and no vertex is repeated.
For a permutation $p = (a_1,a_2,\ldots,a_9)$ of the digits $1,2,\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$. Let $n$ denote the number of permutations $p$ with $s(p) = m$. Find $|m - n|$.
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where $y=0$, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where $y < 0$. Freddy starts his search at the point $(0, 21)$ and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.
Camy made a list of every possible distinct five-digit positive even integer that can be formed using each of the digits 1,3, 4, 5 and 9 exactly once in each integer. What is the sum of the integers on Camy’s list?
There is a $40\%$ chance of rain on Saturday and a $30\%$ chance of rain on Sunday. However, it is twice as likely to rain on Sunday if it rains on Saturday than if it does not rain on Saturday. The probability that it rains at least one day this weekend is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Find $a+b$.
Find the number of sets $\{a,b,c\}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$.
The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with the same color
Beatrix is going to place six rooks on a $6 \times 6$ chessboard where both the rows and columns are labeled $1$ to $6$; the rooks are placed so that no two rooks are in the same row or the same column. The $value$ of a square is the sum of its row number and column number. The $score$ of an arrangement of rooks is the least value of any occupied square. Find the average score over all valid configurations.
How many rectangles of any size are in the grid shown here?
The Beavers, Ducks, Platypuses and Narwhals are the only four basketball teams remaining in a single-elimination tournament. Each round consists of the teams playing in pairs with the winner of each game continuing to the next round. If the teams are randomly paired and each has an equal probability of winning any game, what is the probability that the Ducks and the Beavers will play each other in one of the two rounds? Express your answer as a common fraction.
A spinner is divided into 5 sectors as shown. Each of the central angles of sectors 1 through 3 measures $60^\circ$ while each of the central angles of sectors 4 and 5 measures $90^\circ$. If the spinner is spun twice, what is the probability that at least one spin lands on an even number? Express your answer as a common fraction.
The student council at Round Junior High School has eight members who meet at a circular table. If the four officers must sit together in any order, how many distinguishable circular seating orders are possible? Two seating orders are distinguishable if one is not a rotation of the other.
Consider a coordinate plane with the points $A(−5, 0)$ and $B(5, 0)$. For how many points $X$ in the plane is it true that $XA$ and$XB$ are both positive integer distances, each less than or equal to 10?
A fair coin is flipped four times. Written as a percent, what is the probability of getting two heads and two tails, in any order? Express your answer to the nearest tenth.
Each card in a particular deck of cards contains a number denoting its value from 2 to 6, inclusive. The deck is made up of four cards of each value for
a total of 20 cards. If two of these cards are chosen at random and without replacement, what is the probability that the sum of their values is less than 10? Express your answer as a common fraction.
Eight blue and five orange tiles are arranged in an ordered line such that the tile on the left must be blue and every tile must be adjacent to at least one tile of the same color. For example, if an arrangement of four tiles was made, the only possibilities would be $BBBB$ or $BBOO$. How many different arrangements are possible if all thirteen tiles must be used?
Compute the number of permutations $x_1, \cdots, x_6$ of integers $1, \cdots, 6$ such that $x_{i+1}\le 2x_i$ for all $i, 1\le i < 6$.
An $n$-sided die has the integers between $1$ and $n$ (inclusive) on its faces. All values on the faces of this die are equally likely to be rolled. An $8$-sides side, a $12$-sided die, and a $20$-sided die are rolled. Compute the probability that one of the values rolled equal to the sum of the other two values rolled.
Compute the value of $$\sum_{n=2019}^\infty\frac{1}{\binom{n}{2019}}$$
Show that $$\binom{n}{1}-\frac{1}{2}\binom{n}{2}+\frac{1}{3}\binom{n}{3}-\cdots+(-1)^{n+1}\binom{n}{n}= 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}$$