Find the value of $$\binom{n}{0}-\binom{n}{1}+\binom{n}{2}-\binom{n}{3}+\cdots +(-1)^n\binom{n}{n}$$
John walks from point $A$ to $C$ while Mary goes from point $B$ to $D$. Both of them will move along the grid, either right or up, so they take shortest routes. How many different possibilities are there such that their routes do not intersect?
Show that $$\sum_{k=0}^n\left(2^k\binom{n}{k}\right)=3^n$$
Assuming a small packet of mm’s can contain anywhere from $20$ to $40$ mm’s in $6$ different colours. How many different mm packets are possible?
Show that $$\sum_{k=1}^n \binom{n}{k}\binom{n}{k-1}=\binom{2n}{n-1}$$
Simplify $$\binom{n}{0} - \frac{1}{2}\binom{n}{1} +\binom{n}{2} - \frac{1}{2}\binom{n}{3} + \cdots $$
Let $n > k$ be two positive integers. Simplify the following expression $$\binom{n}{k} + 2\binom{n-1}{k} + 3\binom{n-2}{k} + \cdots+ (n-k+1)\binom{k}{k}$$
Let positive integers $m$ and $n$ satisfy $m\le n$. Prove $$\sum_{k=m}^n\binom{n}{k}\binom{k}{m}=2^{n-m}\binom{n}{m}$$
Show that $$\sum_{k=0}^{2n-1}(-1)^k(k+1)\binom{2n}{k}^{-1}=\frac{1}{\binom{2n}{0}}-\frac{2}{\binom{2n}{1}}+\cdots-\frac{2n}{\binom{2n}{2n-1}}=0$$
Let $(x^{2014} + x^{2016}+2)^{2015}=a_0 + a_1x+\cdots+a_nx^2$, Find the value of $$a_0-\frac{a_1}{2}-\frac{a_2}{2}+a_3-\frac{a_4}{2}-\frac{a_5}{2}+a_6-\cdots$$
Compute $$\binom{2022}{1} - \binom{2022}{3} + \binom{2022}{5}-\cdots + \binom{2022}{2021}$$
While playing table tennis against Jordan, Chad came up with a new way of scoring. After the first
point, the score is regarded as a ratio. Whenever possible, the ratio is reduced to its simplest form. For
example, if Chad scores the first two points of the game, the score is reduced from $2:0$ to $1:0$. If later
in the game Chad has $5$ points and Jordan has $9$, and Chad scores a point, the score is automatically
reduced from $6:9$ to $2:3$. Chad's next point would tie the game at $1:1$. Like normal table tennis, a
player wins if he or she is the first to obtain $21$ points. However, he or she does not win if after his
or her receipt of the $21^{st}$ point, the score is immediately reduced. Chad and Jordan start at $0:0$ and
finish the game using this rule, after which Jordan notes a curiosity: the score was never reduced. How
many possible games could they have played? Two games are considered the same if and only if they
include the exact same sequence of scoring.
Jack wants to bike from his house to Jill's house, which is located three blocks east and two blocks north of Jack's house. After biking each block, Jack can continue either east or north, but he needs to avoid a dangerous intersection one block east and one block north of his house. In how many ways can he reach Jill's house by biking a total of five blocks?
A magazine printed photos of three celebrities along with three photos of the celebrities as babies. The baby pictures did not identify the celebrities. Readers were asked to match each celebrity with the correct baby pictures. What probability that a reader guesses at random will match three
- all correctly?
- all incorrectly?
The "Middle School Eight" basketball conference has $8$ teams. Every season, each team plays every other conference team twice (home and away), and each team also plays $4$ games against non-conference opponents. What is the total number of games in a season involving the "Middle School Eight" teams?
Four children were born at City Hospital yesterday. Assume each child is equally likely to be a boy or a girl. Which of the following outcomes is most likely?
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of $5$ chairs under these conditions?
How many triangles with positive area have all their vertices at points $(i,j)$ in the coordinate plane, where $i$ and $j$ are integers between $1$ and $5$, inclusive?
How many integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $211$ have this property.
A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ so that one side of the square lies on the hypotenuse of the triangle. What is $\tfrac{x}{y}$?
The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
Call a positive integer $monotonous$ if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, $3$, $23578$, and $987620$ are monotonous, but $88$, $7434$, and $23557$ are not. How many monotonous positive integers are there?
A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure $x$. Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is $\frac{14}{25}$. Find the difference between the largest and smallest possible values of $x$.
For nonnegative integers $a$ and $b$ with $a + b \leq 6$, let $T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}$. If $S$ denotes the sum of all $T(a, b)$, where $a$ and $b$ are nonnegative integers with $a + b \leq 6$. Find $S$.
A special deck of cards contains $49$ cards, each labeled with a number from $1$ to $7$ and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, find the probability that Sharon can discard one of her cards and $\textit{still}$ have at least one card of each color and at least one card with each number.