📘 combinatorics

1. **Problem statement:** We have 6 points on a circle and want to find the number of polygons with 3 or 4 sides that can be formed by connecting these points. 2. **Formula and exp

1. **State the problem:** We need to find how many teams of 5 people can be created from 45 people registered for a basketball tournament. 2. **Formula used:** The number of teams

1. **Problem statement:** Calculate the binomial coefficient $\binom{8}{2}$. 2. **Formula:** The binomial coefficient $\binom{n}{k}$ is calculated by the formula:

1. **Problem statement:** We want to find how many different codes can be created with 2 letters, 2 digits, and 2 letters in that order, with no repeated letters or digits, and exa

1. **Problem statement:** (i) Find the number of different seating arrangements of eight friends sitting together in a row.

1. Das Problem besteht darin, alle möglichen Wurfkombinationen beim Dart zu finden, die eine bestimmte Punktzahl ergeben, beginnend mit 3 und dann 4. 2. Wir betrachten die mögliche

1. **Stating the problem:** We need to form a 5-digit number using digits from 0 to 9 without repetition. The number cannot start with zero and must be even.

1. The problem involves understanding factorials and combinations as given by the expressions $S! = 120$, $\frac{12!}{2!2!2!2!} = 28337600$, and $C_5^3 = 10$. 2. First, recognize t

1. **State the problem:** We want to find the number of ways to make 80 cents using only quarters (25 cents), dimes (10 cents), and nickels (5 cents). 2. **Set variables:** Let $q$

1. The problem asks: How many of the same color are there? This question is ambiguous without additional context such as the total number of items or colors involved. 2. To solve p

1. **Problem Statement:** We want to find the number of ways to arrange 8 objects into 3 places. This is a permutation problem because the order matters. 2. **Formula Used:** The n

1. **Problem statement:** Three runners compete in a race. We want to find the number of ways they can finish the race under two conditions: a) No tied places.

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.