📘 combinatorics

1. **State the problem:** We need to find the number of subsets and the number of proper subsets of the set $\{31, 3, 17, 26, 8, 21\}$. The set has 6 elements. 2. **Formula for num

1. **Problem:** Determine how many cards in a 52-card deck fit the description: face cards or black cards. 2. **Definitions and facts:**

1. **State the problem:** We need to find how many different pizzas can be made by choosing one dough, one sauce, and one topping. 2. **Identify the choices:**

1. **Problem Statement:** We have a 5 × 4 grid with a 2 × 2 black square occupying the top-left corner (cells in rows 1-2, columns 1-2). We want to place five 1 × 2 horizontal rect

1. **Problem:** How many ways can we select one white and one black square on a chessboard? And how many ways to select two squares of any color? The chessboard has 64 squares, hal

1. The question "are there any other possible combinations" is quite general and needs context to solve mathematically. 2. Combinations refer to the number of ways to select items

1. 题目说明：有5名同学A、B、C、D、E排成一排照相，要求A、B、C中至少有两个人相邻。 2. 计算总排列数：5个人全排列数为$$5! = 120$$。

1. 问题陈述： 我们要解释组合数学中符号 $A_{3,2}$ 中的数字 "3" 和 "2" 分别代表什么。

1. **State the problem:** We want to find the number of ways to arrange 4 different books on a shelf. 2. **Formula used:** The number of ways to arrange $n$ different items in orde

1. **State the problem:** Calculate the combination $nCr$ for $n=5$ and $r=3$ using the formula for combinations. 2. **Formula:** The number of combinations of $n$ items taken $r$

1. **Stating the problem:** We need to select a head boy, two deputy head boys, a head girl, and three deputy head girls from a student council of 14 girls and 16 boys. 2. **Unders

1. **Problem statement:** We are given a set of 9 children participating in 3 tasks. 5 children solved the first task, 6 solved the second, and 7 solved the third. Every child solv

1. **Problem statement:** We need to form a group of 5 people from 3 married couples (6 people), 4 male staff, and 5 female staff, with the condition that no married couple is incl

1. **Problem statement:** Determine the number of pathways from point A to point B on a grid where you can only move right or down. 2. **Understanding the grid:** The grid has 3 ro

1. Statement of the problem. After each round of a knock-out netball competition the losing teams drop out until just the winner remains.

1. The problem is to find the number of ways to choose 10 items from a set of $n$ items, which is a combination problem. 2. The formula for combinations is given by:

1. The problem is to generate 60 choose questions, which typically involve combinations. 2. The formula for combinations is given by $$C(n, k) = \frac{n!}{k!(n-k)!}$$ where $n$ is

1. **Problem statement:** We need to find the number of ways to divide 17 chapters among four writers such that the first and third writers write 5 chapters each, the second writer

1. The problem asks to find a trio (group of three digits) from the number 123456 without repeating any pair of digits. 2. We interpret "without repeating a pair" as selecting thre

1. The problem is to create a trio (group of three numbers) from the digits 1, 2, 3, 4, 5, 6 without repeating any sequence. 2. We interpret "without repeating a sequence" as formi

1. **Problem:** How many different 4-digit even numbers can be formed from the digits 1, 3, 5, 6, 8, and 9 if no repetition of digits is allowed? 2. **Formula and rules:**