📘 combinatorics

1. **Problem statement:** We want to find the number of ways to arrange 7 different books on a shelf such that two particular books are placed at the ends. 2. **Understanding the p

1. **Problem statement:** We need to find the number of 6-letter passwords using distinct letters from the English alphabet where the first letter must be a vowel. 2. **Important i

1. The problem asks which table lists all the different ways Harry can travel to Hogwarts and back, with each row representing one outcome. 2. To solve this, we consider the possib

1. **Problem Statement:** You have 3 colors (Green, Orange, Yellow) and 2 sizes (Large, Small) of water balloons. You want to list all possible outcomes when picking a color and si

1. **State the problem:** Edwin wants to buy three vehicles: a car, a truck, and a motorcycle. Each vehicle has a set of color options. 2. **Identify the options:**

1. **Problem statement:** Mrs. Rivera has 10 newest gowns and 5 mannequins. She wants to display 5 gowns at a time and change the set every 2 days. We need to find how many days wi

1. **Problem statement:** We need to find the number of ways to select 2 biology books from 7 and 2 chemistry books from 6. 2. **Formula used:** The number of ways to choose $k$ it

1. **Problem statement:** We have a box with 5 red balls, 7 green balls, and 6 yellow balls. We want to find the number of ways to choose 6 balls such that exactly 2 balls of each

1. **Problem statement:** We have digits 2, 4, 5, 7, and 9 and want to find: - Total numerals formed using all digits without repetition.

1. **State the problem:** We need to find the number of different voicemail passwords possible where each password consists of 1 letter followed by a 3-digit number less than 600.

1. The problem asks to solve the combination equations: 2. First, recall the formula for combinations:

1. **Problem Statement:** We have $n$ chicks belonging to $\log n$ hens. Each chick has exactly one mother hen, and a hen can have between 0 and $n$ chicks. We want to find the low

1. **Problem Statement:** We need to state and prove the necessary and sufficient condition for two graphs to be isomorphic and illustrate with examples. 2. **Definition of Graph I

1. **Problem statement:** We have a box with 5 red, 4 blue, and 3 white balls. We want to find the number of ways to select 3 balls such that each ball is a different color. 2. **U

1. **State the problem:** We want to find how many different sundaes can be made by choosing 3 ice cream flavors out of 31, 3 sauces out of 7, and 3 toppings out of 10. 2. **Formul

1. **Problem Statement:** We need to find the total number of distinct signals that can be created by arranging 3 pink, 3 white, and 2 black flags in a straight line. 2. **Formula

1. **Problem Statement:** Show that the sequence defined by $$D_n$$ satisfies the recursive formula $$D_n = nD_{n-1} + (-1)^n$$ for $$n \geq 1$$.

1. The problem asks for the number of 4-digit PIN codes with no repetition of numbers. 2. To find the number of possible 4-digit PIN codes with no repeated digits, we use permutati

1. Ստացեք խնդիրը. Դպրոցում կա 3 փոխտնօրեն և 9 մաթեմատիկայի ուսուցիչ: Պետք է կազմել հանձնաժողով, որը բաղկացած կլինի 1 փոխտնօրենից և 3 ուսուցիչներից: 2. Օգտագործեք համակցությունների

1. **Problem statement:** We need to find the number of ways to arrange 5 different mathematics books and 4 different physics books on a shelf such that no two physics books are ad

1. **Problem statement:** We have 5 men and 4 women to be seated in a row such that no two women sit together. We need to find the number of ways to arrange them under this conditi