📘 combinatorics

1. **Problem statement:** We have a committee of 6 people sitting in a row. Two specific members refuse to sit next to each other. We need to find the number of possible seating or

1. **Problem Statement:** Find the number of partitions of $n=7$ into (i) odd summands and (ii) even summands using generating functions. Then verify by listing all partitions. 2.

1. **Problem Statement:** Find the number of partitions of $n=7$ into (i) odd summands and (ii) even summands using generating functions. Then verify by listing all partitions. 2.

1. **Problem Statement:** Alice has a safe secured by a 4-digit code with no repeated digits. We want to find the number of possible codes that satisfy this condition. 2. **Formula

1. The problem is to find the total number of rugby matches played if each team plays every other team exactly once. 2. The formula to calculate the number of matches in a round-ro

1. **Problem statement:** Helen wants to choose one main course and one dessert from 6 main courses and 8 desserts. 2. **Formula for combinations:** When choosing one item from eac

1. **Problem Statement:** We want to find for how many positive integers $n$ from 1 to 2008, the set $\{1,2,3,\ldots,4n\}$ can be partitioned into $n$ disjoint subsets each contain

1. 問題陳述： 計算一個由4行5列小長方形組成的矩形網格中，總共有多少個長方形。

1. **Stating the problem:** Calculate the permutation $18P17$, which represents the number of ways to arrange 17 objects out of 18 distinct objects. 2. **Formula used:** The permut

1. **Problem statement:** We have 2 different meat tortillas and 4 different vegetable tortillas to arrange on a plate. We want to find the number of ways to arrange them under dif

1. **Stating the problem:** We want to arrange 4 textbooks, 3 exercise books, and 2 manuals on a shelf. We need to find the number of ways to do this under different conditions:

1. **Problem statement:** We have 10 athletes standing in a row for a photo, and the tallest athlete must be positioned exactly in the center. 2. **Understanding the problem:** Sin

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.