📘 combinatorics

1. **Stating the problem:** We need to find the number of different 5-digit combinations possible on a lock with 5 wheels labeled 0 to 9 when successive digits must be different. 2

1. **Stating the problem:** We need to find the number of different 5-digit combinations possible on a lock with 5 wheels labeled 0 to 9 when digits can be repeated. 2. **Understan

1. **Stating the problem:** We need to find the number of different 5-digit combinations possible on a lock with 5 wheels labeled 0 to 9 under different conditions. 2. **Understand

1. **Problem Statement:** We need to find how many 2-character passwords are possible using the English alphabet with case sensitivity. 2. **Understanding the problem:** The Englis

1. **Problem statement:** How many 2-letter code words can be formed from the letters X, N, I, M, A, P under three conditions: - No letter is repeated.

1. **Problem:** How many different combinations of six numbers can be chosen from the digits 1 to 8 if each digit is chosen only once? 2. **Formula:** The number of combinations of

1. **State the problem:** Maeve’s team plays 5 matches, and we want to find the total number of different possible outcomes for these matches. 2. **Understand the outcomes:** Each

1. **Stating the problem:** We have 5 class prefects: Eimear (E), Samantha (S), Mia (M) who are girls, and Paul (P), Neil (N) who are boys. We need to find all possible pairs of on

1. **Problem statement:** (a)(i) A restaurant offers a three-course meal with 4 starters, 6 main courses, and 8 desserts. We need to find the total number of different three-course

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.