📘 combinatorics

1. **State the problem:** John has three types of activities to choose from for his evening: reading a book, watching a video, or going to the movies. 2. **Given data:**

1. **State the problem:** A student must select a two-unit study, choosing one unit from semester one and one unit from semester two. 2. **Identify choices in semester one:** There

1. **Problem statement:** We need to find the number of distinct paths from node A (top-left corner) to node B (bottom-right corner) on a 3x3 grid network. 2. **Allowed moves:** Up

1. **Problem statement:** We have a 4x4 wall (16 squares) painted with two colours: yellow and another colour. At least one diagonal is painted yellow, and the rest of the squares

1. **Problem statement:** We have 9 binary vectors of length 8, and we consider their 36 mutual sums modulo 2 (i.e., sums of pairs of distinct vectors, component-wise mod 2). We wa

1. The problem is to find the value of the binomial coefficient $\binom{2}{5}$. 2. The binomial coefficient $\binom{n}{k}$ is defined as the number of ways to choose $k$ elements f

1. **Problem Statement:** We need to find the number of different ways to select a panel of 12 jurors and 2 alternate jurors from a group of 27 potential jurors. 2. **Understanding

1. **Problem Statement:** We need to find the number of ways to arrange four items from three different departments in a one-page advertisement with 3 rows and 4 columns, such that

1. **Problem statement:** Find the number of ways 4 boys and 4 girls can sit at a square table with two seats on each side such that each side has exactly one boy and one girl. 2.

1. **Stating the problem:** We have a symphony orchestra with 30 Haydn symphonies, 15 modern works, and 9 Beethoven symphonies. The program consists of one Haydn symphony, followed

1. **Problem statement:** We have 12 members (6 married couples) in a committee.

1. **Problem statement:** There are 10 members sitting around a round table, including 1 chairman seat. Among them, 4 members form a written report subcommittee. We want to find th

1. **State the problem:** A spinner with eight sections labeled A through H is spun twice. We need to find the total number of possible outcomes. 2. **Understand the problem:** Eac

1. **State the problem:** We want to find how many 5-letter arrangements can be made from the 26 letters of the English alphabet with no repeated letters. 2. **Formula used:** The

1. **Énoncé du problème :** Montrer l'identité de Vandermonde suivante :

1. The problem is to find the number of combinations of 3 items chosen from a set of $n$ items. 2. The formula for combinations is given by:

1. **Problem statement:** Calculate the number of different serial numbers possible on a dollar bill where the serial number consists of a letter, followed by eight digits, and the

1. **State the problem:** We have a class of 30 kids and want to select 4 of them to write their exam in the library. We need to find how many ways this selection can be done. 2. *

1. Problem: Find the number of different three-digit numbers under various conditions. 2. Formula and rules: For counting numbers with digits, use permutations and combinations.

1. The problem asks to prove a statement involving nonnegative integers $n$ and $r$. 2. Since the exact statement to prove is missing, let's assume it involves a common combinatori

1. **Problem:** Find the number of ways to seat 8 choristers around a round table such that two particular choristers must sit together. 2. **Formula and Explanation:** When two pa