Meena writes the numbers $1$, $2$, $3$, and $4$ in some order on a blackboard, such that she cannot swap two numbers and obtain the sequence $1$, $2$, $3$, $4$. How many sequences could she have written?
How many length ten strings consisting of only $A$s and $B$s contain neither "$BAB$" nor "$BBB$" as a substring?
How many different ways to express $13$ as the sum of several positive odd numbers? Order
matters. For example, $1 + 1 + 3 + 3 + 5$ is treated as a different expression as 1 + 3 + 1 + 3 + 5
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe.
25 boys and 8 girls sit in a circle. If there are at least two boys between any two girls, how many different sitting plans are there? (Two sitting plans will be considered as the same if they differ just by rotating.)
Joe is playing with a set of $6$ masses: $1$g, $2$g, $4$g, $8$g, $16$g, and $32$g. He found that some weights can be measured in more than one way. For example, $7$g can be measure by putting $1$g, $2$g, and $4$g on one side of a balance. It can also be achieved by putting $1$g and $8$g on different sides of a balance. He therefore wonder which weight can be measured using these masses in the most number of different ways? Can you help him to find it out? Describe how will you approach this problem. The final answer is optional.
Cozy the Cat is going up a staircase of $10$ steps. She can either walk up $1$ step a time or jump $2$ steps a time. How many different ways can she reach the top of this staircase?
How many different $6$-digit numbers can be formed by using digits $1$, $2$, and $3$, if no adjacent digits can be the same?
Joe wants to write $1$ to $n$ in a $1 \times n$ grid. The number 1 can be written in any grid, while the number $2$ must be written next to $1$ (can be at either side) so that these two numbers are together. The number 3 must be written next to this two-number block. This process goes on. Every new number written must stay next to the existing number block. How many different ways can Joe fill this $1 \times n$ grid?
How many different ways are there to cover a $1\times 10$ grid with some $1\times 1$ and $1\times 2$ pieces without overlapping?
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?
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
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 $\log_2 n$.
A word is an ordered, non-empty sequence of letters, such as word or wrod. How many distinct words can be made from a subset of the letters $\textit{c, o, m, b, o}$, where each letter in the list is used no more than the number of times it appears?