📘 combinatorics

1. **State the problem:** We need to find the number of possible license plates with the format of three digits followed by three letters. 2. **Given information:**

1. **State the problem:** Evaluate the expression $$\frac{C(9,5) \cdot C(13,9)}{C(12,11)}$$ where $C(n,k)$ is the binomial coefficient representing combinations. 2. **Recall the fo

1. **State the problem:** You want to find how many different smoothies you can make using 3 types of fruit, 4 types of veggies, and 2 types of juice. 2. **Formula used:** When cho

1. **صورت مسئله:** در یک سالن تئاتر ۵ درب وجود دارد. اگر شخصی از درب A وارد شود، باید از درب دیگری خارج شود. می‌خواهیم بدانیم مریم چند روش مختلف برای ورود و سپس خروج دارد.

1. **صورت مسئله:** از حروف کلمهٔ «صندلی» می‌خواهیم چند کلمهٔ ۵ حرفی بدون تکرار بسازیم، به‌طوری که فقط **یک** حرفِ نقطه‌دار در کلمهٔ حاصل باشد. 2. **حروف کلمهٔ «صندلی» را بررسی می‌ک

1. **صورت مسئله:** می‌خواهیم بدانیم به چند طریق می‌توان به $8$ سوال $3$ گزینه‌ای پاسخ داد، به شرطی که حتما به $4$ سوال اول پاسخ داده شود. 2. **فرمول مورد استفاده:** اگر هر سوال $3$

1. The problem asks how many themes, locations, and days Farah is considering for her birthday party based on the tree diagram. 2. From the tree diagram, we see the branches repres

1. **Problem statement:** We have a deck of cards with colors and numbers, each card worth $2^m$ points where $m$ is the card's number. - Original deck: 10 red (1 to 10), 10 blue (

1. **Problem statement:** We have a $32 \times 32 \times 32$ cube made of unit cubes, all six faces painted. We consider all subcubes inside it (of any side length from 1 to 32). W

1. **State the problem:** There are 10 football teams in a league, and each team plays every other team 4 times. We need to find the total number of games played. 2. **Formula used

1. **Problem restatement:** You provided counts of 3-card combinations from a 32-card deck with point totals and example card combinations.

1. **Problem statement:** We have a card game with 32 cards, and each hand consists of 3 cards of the same suit.

1. **Probleem 1: Hatice gooit met zeven dobbelstenen. Hoeveel mogelijkheden zijn er om in totaal 40 ogen te gooien?** 2. **Formule en regels**

1. The problem is to evaluate the binomial coefficient $\binom{3}{}$, which is incomplete as it lacks the second parameter. 2. The binomial coefficient $\binom{n}{k}$ represents th

1. **State the problem:** We need to find the number of ways to arrange 20 cars in a row where the cars are distinguishable only by their colors. There are 4 blue, 3 black, 5 yello

1. **State the problem:** We want to find how many different five-letter codes can be made using the letters \(a, b, c, d, e\). Then, we want to find how many of these codes start

1. **Problem Statement:** We have ID numbers starting with letter S, followed by 2 digits for the year, then 6 digits.

1. **Stating the problem:** We have a menu with 4 entrée options (Burger, Taco, Rice, Spaghetti), 2 snack options (Chips, Apple), and 2 drink options (Water, Milk).

1. **Problem statement:** We have two girls (Arta, Besa) and three boys (Dioni, Ermiri, Flamur) who want to sit in a row of five seats such that the girls sit next to each other an

1. **Stating the problem:** Dominik has a 3×3 grid with a 2×2 shaded block in the bottom right corner. He wants to place a dark and a light stone so that they are not in the same r

1. **Stating the problem:** We need to find how many different routes can be chosen following the indicated arrow directions while avoiding the black stones. 2. **Understanding the