Practice (68)
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Two people agree to meet at a place some time in the next 10 days. They have also agreed whoever arrives the place should wait for the other for 3 days and then leave. What is the probability that they will see each other?
In the following diagram, $\overline{AO}= 2$, $\overline{BO} = 5$, and $\angle{AOB} = 60^\circ$. Point $C$ is selected on $\overline{BO}$ randomly. Find the probability that $\triangle{AOC}$ is an acute triangle.
How many triangles are there in the following diagram?
How many rectangles or squares are there in the following diagram?
Six people form a line. $A$ must stand after $B$ (not necessarily immediately after $B$). How many different ways are there to form such a line?
Seven people form a line. If $A$ must stand next to $B$, and $C$ must stand next to $D$, how many possibilities are there?
Team MAS won a total of $10$ gold medals in a $6$-day tournament. It won at least one gold medal every day. How many different possibilities are there to count the number of gold medals won each day?
Find the number of positive integer solutions to the following equation: $$x_1+x_2+\cdots+x_5=14$$
Joe goes to a supermarket to buy 10 cakes. There are 6 different types of cakes, and each type has a sufficient quantity. How many different combinations of cakes can Joe have?
Library MAS has m bookshelves for books of m different categories. Each bookshelf has n books. Joe, the librarian, needs to re-arrange these books. Books of the same category still need to be put on the same bookshelf, but their order can change.
How many different arrangement plans are there that:
(a) no book is put on the same bookshelf as before?
(b) no book is put on it original position?
(c) no book is on the same bookshelf, and no book is at the relative position in the new bookshelf as it was before
Let $D_n$ be the derangement count, prove:
- $D_n =n\cdot D_{n−1} +(−1)^n$
- $D_n = (n−1)\cdot (D_{n−2} +D_{n−1})$
Joe has sufficient quantity of pennies, nickels, dimes, and quarters. He wants to pay a $\$1$ bill using these coins. How many different combinations does he have?
How many different weights can be measured using a set of 4 masses of 1, 2, 3, 4 grams each? For each measurable weight, how many different ways are possible?
How many different ways to express 13 as the sum of some positive odd integers? These integers do not need to be unique. Sequence of these integers also matters. For example $5 + 7 + 1$ and $7 + 1 + 5$ will be treated as two different ways.
How many different ways are there to express $20$ as the sum of $1$, $2$, and $5$? (All numbers must appear.)
There are $2$ white balls, $3$ red balls, and $1$ yellow ball in a jar. How many different ways are there to retrieve $3$ balls?
There are $2$ white balls, $3$ red balls, and $1$ yellow balls in a jar. How many different ways are there to retrieve $3$ balls to form a line?
How many different $5$-digit numbers can be formed using $1$, $2$, $3$, and $4$ that satisfy the following conditions:
- the digit $1$ must appear either $2$ or $3$ times,
- the digit $2$ must appear even times,
- the digit $3$ must appear odd times, and
- the digit $4$ has no restriction
There are $60$ friends who want to visit each others home during summer vacation. Everyday, they decide to either stay home or visit the home of everyone who stayed home that day. Find the minimum number of days required for everyone to have visited their friends' homes.
A girl and a guy are going to arrive at a train station. If they arrive within $10$ minutes of each other, they will instantly fall in love and live happily ever after. But after $10$ minutes, whichever one arrives first will fall asleep and they will be forever alone. The girl will arrive between $8$ AM and $9$ AM with equal probability. The guy will arrive between $7$ AM and $8:30$ AM, also with equal probability. Find the probability that the probability that they fall in love.
Consider an orange and black coloring of a $20\times 14$ square grid. Let $n$ be the number of coloring such that every row and column has an even number of orange square. Evaluate $\log_2 n$.
For all positive integer $n$, show that $$\sum_{k=1}^n\frac{k\cdot k! \cdot\binom{n}{k}}{n^k}=n$$
The Centerville Middle School chess team consists of two boys and three girls. A photographer wants to take a picture of the team to appear in the local newspaper. She decides to have them sit in a row with a boy at each end and the three girls in the middle. How many such arrangements are possible?
Each of two boxes contains three chips numbered $1$, $2$, $3$. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. What is the probability that their product is even?
How many integers between $1000$ and $9999$ have four distinct digits?