📘 combinatorics

### Problème 1a 1. **Énoncé :** Un livre a 12 pages, chaque page est coupée en 3 parties (têtes, troncs, jambes). On peut mélanger ces parties pour former des personnages différent

1. **Постановка задачі:** Маємо смугу довжиною 100 одиничних квадратів, у які послідовно записують числа від $1$ до $100$. Число $1$ записується у будь-який квадрат, а кожне наступ

1. **State the problem:** We need to find the number of ways to place two rooks on an 8x8 chessboard so that they do not attack each other. 2. **Important rule:** Rooks attack alon

1. **Stating the problem:** Frank can choose a combo menu with one type of popcorn, one type of soda, and one type of snack. We want to find out how many ways he can do this.

1. **Stating the problem:** We have 16 volleyball teams divided into 3 ranking tiers: 4 teams in 1st tier, 4 teams in 2nd tier, and 8 teams in 3rd tier.

1. **Problem:** How many ways can you make a 4-topping pizza if there are 12 toppings to choose from? 2. **Formula:** This is a combination problem because the order of toppings do

1. **Stating the problem:** Andi has 15 identical marbles and wants to put them into 3 containers such that no container is empty. We need to find the number of ways to do this.

1. مسئله: با ارقام ۰، ۱، ۵، ۷، ۸ و بدون تکرار رقم، چند عدد سه‌رقمی فرد می‌توان نوشت؟ 2. تعریف عدد سه‌رقمی فرد: عددی که سه رقم دارد و رقم یکان آن فرد باشد.

1. **Problem:** A board of directors has 10 members. We want to elect a president, a vice president, and a secretary in that order. How many ways can this be done? 2. **Formula and

1. **Problem 14:** Jan grabs a pair of slippers from 4 people (Joe, Jan, Jen, Jun), each with a pair of slippers. We want the probability that Jan did *not* grab a matching pair. 2

1. **பிரச்சினை விளக்கம்:** 5 தூப்பாட்ட மடைகள், 4 நந்துகள் மற்றும் 3 கையுறைகள் 6 மாணவர்களுக்கு பகிரப்படுகின்றன.

1. **State the problem:** Evaluate the combination expression $11 C 2$. 2. **Formula:** The combination formula is given by

1. The problem asks to evaluate the combination $14C5$, which represents the number of ways to choose 5 items from 14 without regard to order. 2. The formula for combinations is:

1. The problem asks to evaluate the combination expression $15C4$, which represents the number of ways to choose 4 items from 15 without regard to order. 2. The formula for combina

1. **Problem statement:** (a) Find how many 3-digit and 4-digit numbers can be formed using digits 4, 5, 6, 7, 8 without repetition.

1. The problem is to evaluate the binomial coefficient $\binom{2}{2}$. 2. The binomial coefficient $\binom{n}{k}$ is defined as:

1. **Problem Statement:** (A) How many ways can a 4-person subcommittee be selected from a committee of 8 people?

1. **State the problem:** We want to find how many different possible finishes there are for the first 3 places in a horse race with 18 horses, excluding ties. 2. **Formula used:**

1. The problem asks to evaluate the combination $7 C 3$, which represents the number of ways to choose 3 items from 7 without regard to order. 2. The formula for combinations is:

1. **Problem statement:** Find the number of license plates with 4 letters followed by 2 digits where no letters or digits are repeated. 2. **Formula and rules:**

1. **Problem statement:** We want to find the number of different license plates possible with 4 letters followed by 2 digits. 2. **Total possible license plates:** Each of the 4 l