Recent questions tagged combinatory

How many 15 -letter arrangements of 5 A's, 5 B's, and 5 C's have no A's in the first 5 letters, no B's in the next 5 letters, and no C's in the last 5 letters?$\sum_{k=0}...

A large equilateral triangle is constructed by using toothpicks to create rows of small equilateral triangles. For example, in the figure, we have 3 rows of small congrue...

Pat is to select six cookies from a tray containing only chocolate chip, oatmeal, and peanut butter cookies. There are at least six of each of these three kinds of cookie...

Let the set $\mathcal{S}=\{8,5,1,13,34,3,21,2\}$. Susan makes a list as follows: for each two-element subset of $\mathcal{S}$, she writes on her list the greater of the s...

How many three-digit numbers have at least one 2 and at least one 3 ?52545658

A librarian is arranging $4$ mystery novels, $3$ history books, $2$ poetry books, and $2$ travel books on a shelf. All books are distinct. How many ways can the books be ...

During the GO Classes Jaipur felicitation trip, Deepak Sir and Sachin Sir are making two separate cricket teams from the toppers. There are $6$ batters and $6$ bowlers av...

The English alphabet has $26$ letters, of which $5$ are vowels: $A,E,I,O,U$. How many $8$-letter strings, with all letters distinct, can be formed that start with $B$, en...

At an academy, there are $30$ students eligible for a $10$-student research team. There are $8$ students in Club $A$, $7$ students in Club $B$, and exactly $3$ students a...

A museum has two sealed collections of artifacts. Collection $X$ contains $5$ paintings, $4$ sculptures, and $3$ manuscripts. Collection $Y$ contains $6$ paintings, $5$ s...

There are 1001 red marbles and 1001 black marbles in a box. Let $P_s$ be the probability that two marbles drawn at random from the box are the same color, and let $P_d$ b...

Juan rolls a fair regular octahedral die marked with the numbers 1 through 8 . Then Amal rolls a fair six-sided die. What is the probability that the product of the two r...

Four distinct circles are drawn in a plane. What is the maximum number of points where at least two of the circles intersect?891012

How many different integers can be expressed as the sum of three distinct members of the set $\{1,4,7,10,13,16,19\}$ ?13162430

According to the standard convention for exponentiation,$$2^{2^{2^2}}=2^{\left(2^{\left(2^2\right)}\right)}=2^{16}=65536 .$$If the order in which the exponentiations are ...

Tina randomly selects two distinct numbers from the set $\{1,2,3,4,5\}$, and Sergio randomly selects a number from the set $\{1,2, \ldots, 10\}$. What is the probability ...

Mary typed a six-digit number, but the two 1 s she typed didn't show. What appeared was 2002 . How many different six-digit numbers could she have typed?481015

What is the smallest integer $n$ for which any subset of $\{1,2,3, \ldots, 20\}$ of size $n$ must contain two numbers that differ by 8 ?281213

Using the letters $A, M, O, S$, and $U$, we can form five-letter "words". If these "words" are arranged in alphabetical order, then the "word" $U S A M O$ occupies positi...

Points $R, S$ and $T$ are vertices of an equilateral triangle, and points $X, Y$ and $Z$ are midpoints of its sides. How many noncongruent triangles can be drawn using an...

A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the whit...

Pat wants to buy four donuts from an ample supply of three types of donuts: glazed, chocolate, and powdered. How many different selections are possible?691215

The Fibonacci sequence $1,1,2,3,5,8,13,21, \ldots$ starts with two 1 s, and each term afterwards is the sum of its two predecessors. Which one of the ten digits is the la...

Professor Gamble buys a lottery ticket, which requires that he pick six different integers from 1 through 46 , inclusive. He chooses his numbers so that the sum of the ba...

There are 5 yellow pegs, 4 red pegs, 3 green pegs, 2 blue pegs, and 1 orange peg to be placed on a triangular peg board. In how many ways can the pegs be placed so that n...

Six points on a circle are given. Four of the chords joining pairs of the six points are selected at random. What is the probability that the four chords form a convex qu...

Call a 7 -digit telephone number $d_1 d_2 d_3-d_4 d_5 d_6 d_7$ memorable if the prefix sequence $d_1 d_2 d_3$ is exactly the same as either of the sequences $d_4 d_5 d_6$...

Color the surfaces of a cube of dimension $5 \times 5 \times 5$ red, and then cut the cube into smaller cubes of dimension $1 \times 1 \times 1$. Take out all the smaller...

How many paths from $A$ to $B$ consist of exactly six line segments (vertical, horizontal or inclined)?

In a die, 1 and 6,2 and 5,3 and 4 appear on opposite faces. When 2 dice are thrown, product of numbers appearing on the top and bottom faces of the 2 dice are formed as f...