📘 combinatorics

1. **Problem statement:** A student must answer 10 out of 13 questions. (i) He must answer at least the first two from the first 5 questions.

1. The problem is to find the value of $C(8,5)$, which represents the number of combinations of 8 items taken 5 at a time. 2. The formula for combinations is:

1. **Problem statement:** We have an alphabet $\Omega = \{X, Y, Z, T, 0, 1, 2, 3, 4, 5, 6, 7\}$ with 12 characters, where 4 are letters ($X,Y,Z,T$) and 8 are digits ($0$ to $7$). W

1. **Problem statement:** We have an alphabet $\Omega = \{X, Y, Z, T, 0, 1, 2, 3, 4, 5, 6, 7\}$ with 12 characters: 4 letters ($X,Y,Z,T$) and 8 digits ($0$ to $7$). A Q-key of leng

1. **State the problem:** We need to find how many different committees of 3 men and 4 women can be formed from 8 men and 6 women. 2. **Formula used:** The number of ways to choose

1. **State the problem:** We want to find the number of ways to select 6 questions out of 10. 2. **Formula used:** The number of ways to choose $k$ items from $n$ items without reg

1. **Stating the problem:** Given the equation $P(n, 2) = C(n + 1, 3)$, find the value of $n$ that satisfies this.

1. **Stating the problem:** Given the equation $P(n, 2) = C(n + 1, 3)$, find the value of $n$ that satisfies this. 2. **Recall formulas:**

1. **State the problem:** There are 10 contestants in a speech contest, and we want to find how many possible ways the first, second, and third presenters can be chosen from these

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