📘 combinatorics

1. **Problem Statement:** An English teacher wants to select 10 books from 5 novels, 6 plays, 6 poetry books, and 8 nonfiction books, with no more than 2 nonfiction books included.

1. The problem involves the binomial coefficient notation $\binom{n}{k}$, which represents the number of ways to choose $k$ elements from a set of $n$ elements. 2. The formula for

1. The problem involves the binomial coefficient notation $\binom{n}{k}$, which represents the number of ways to choose $k$ elements from a set of $n$ elements. 2. The formula for

1. The problem involves understanding the binomial coefficient notation $\binom{n}{k}$, which represents the number of ways to choose $k$ elements from a set of $n$ elements. 2. Th

1. **Problem Statement:** We have $10$ students with the following skills:

1. **Problem 1:** Find the number of permutations where 6 men and 2 women stand in a line with the 2 women next to each other. 2. **Formula and explanation:** Treat the 2 women as

1. **State the problem:** We need to find how many ways Ms. Lam can arrange 3 sopranos out of 10 in the first row. 2. **Formula used:** Since the order of arrangement matters (arra

1. **State the problem:** We want to find how many different combinations of 3 toppings can be chosen from 10 available toppings. 2. **Formula used:** The number of combinations of

1. **State the problem:** We want to find the number of ways to arrange 8 teams in the first, second, and third positions without any ties. 2. **Formula used:** This is a permutati

1. **Problem statement:** We want to find the percentage probability that each of the 4 friends is on a different team when 25 players (4 friends + 21 others) are divided into 5 te

1. **Problem Statement:** We have a grid $a$ with $n$ rows and $m$ columns, each cell $(i,j)$ containing a non-negative integer $a_{i,j}$. We define a path $p$ of length $k$ as a s

1. **Énoncé du problème** Nous devons déterminer le nombre d'assemblages différents des lettres du mot EMPOISONNEMENT dans trois cas :

1. **State the problem:** We need to find the total number of ways a woman can be attired if she wears exactly one hat, one blouse, one skirt, one handbag, and one pair of shoes. 2

1. **Đặt vấn đề:** Chúng ta cần đếm số cách đi từ nút $A$ đến nút $B$ trên một lưới hình chữ nhật với các quy tắc di chuyển:

1. **Problem statement:** We have two parts:

1. ปัญหานี้ถามหาจำนวนวิธีที่สมชายสามารถเลือกสวมเสื้อและกางเกงไปเที่ยวได้ 2. สูตรที่ใช้คือการคูณจำนวนตัวเลือกของเสื้อกับจำนวนตัวเลือกของกางเกง เพราะแต่ละตัวเลือกของเสื้อสามารถจับคู่

1. **Đặt vấn đề:** Chúng ta có hai hộp bi:

1. **Đặt vấn đề:** Tính tổng số cách chọn $19$ viên bi từ $2$ hộp sao cho có ít nhất $17$ viên bi màu đỏ.

1. **Problem statement:** Given a positive integer $n$, find the number of strictly increasing sequences of positive integers whose sum is $n$. 2. **Understanding the problem:** We

1. **State the problem:** We have $14$ students and want to form a committee of $4$ people: $1$ chairperson, $1$ vice-chairperson, and $2$ members. 2. **Understand the roles:** The

1. **Memahami masalah:** Diberikan 3 siswa kelas X, 4 siswa kelas XI, dan 5 siswa kelas XII.