📘 combinatorics

1. সমস্যাটি হলো: আমাদের কাছে ৪ রঙের বল আছে - লাল ৫০টি, সবুজ ৪০টি, নীল ৩০টি, কালো ২০টি। আমাদের লক্ষ্য হলো একই রঙের অন্তত ১০টি বল নিশ্চিতভাবে পাওয়ার জন্য সর্বনিম্ন কতগুলি বল তুলতে হ

1. **Problem Statement:** Find the number of words that can be formed with the letters of the word 'BANANARAMA' without changing the relative position of vowels and consonants. 2.

1. **Problem statement:** We have 5 couples (10 people) to be seated in a row. We want to find the number of ways to arrange them under different conditions. 2. **General formula f

1. **Problem Statement:** We want to find how many ways the digits of 123123123 can be arranged so that the resulting number contains the sequence 123 or the sequence 321. 2. **Und

1. **Problem Statement:** Find the number of ways to arrange the digits of 111222333 such that the resulting number contains the sequence 123 or 321. 2. **Given:** The digits are 1

1. **Problem statement:** We have 5 people (A, B, C, D, E) and 5 chairs in a row. We want to find the number of seating arrangements where A and B are **not** sitting next to each

1. مسئله: چند کلمه 7 حرفی می‌توان ساخت با حروف A, B, C, D, D, E, E به طوری که حروف A, B, C کنار هم نباشند؟ 2. ابتدا تعداد کل کلمات ممکن را بدون هیچ محدودیتی محاسبه می‌کنیم. چون حرو

1. **Problem statement:** Find the number of ways to seat five couples in a row under different conditions. 2. **General formula:** The number of ways to arrange $n$ distinct peopl

1. Problem: Find the number of ways to arrange 2 couples in a row under different restrictions. 2. Formula: The number of permutations of $n$ distinct objects is given by $$n!$$.

1. **Problem:** In a 10-item Mathematics problem-solving test, how many ways can you select 5 problems to solve? 2. **Formula:** The number of ways to choose $k$ items from $n$ ite

1. Problem: How many 5-member committees can be formed from 9 sophomores and 12 seniors under different conditions? 2. Formula: The number of ways to choose $k$ members from $n$ is

1. **Problem statement:** Find the number of ways to arrange the letters of the word INFORMATION such that all vowels are together. 2. **Identify vowels and consonants:** The word

1. Masalah: Berapa banyak susunan kata yang dapat dibentuk dari kata PRASMANAN? 2. Kata PRASMANAN memiliki 9 huruf dengan pengulangan huruf: A sebanyak 2 kali, N sebanyak 2 kali.

1. **Problem statement:** Consider a circle with $n$ points marked on it, where $n \geq 2$. We want to find how many different chords can be drawn by connecting two of these $n$ po

1. **Problem statement:** There are 4 different Mathematics books and 5 different Filipino books. We want to find the number of ways to arrange these books on a shelf such that boo

1. **Problem:** There are 4 different Mathematics books and 5 different Filipino books. In how many ways can the books be arranged on a shelf if books of the same subject must be p

1. **State the problem:** We need to find how many different ways to select 66 numbers from 63 numbers (1 through 63) where order does not matter. 2. **Formula used:** The number o

1. **Problem:** Find the number of permutations of 7 different objects taken 4 at a time without repetition. 2. **Formula:** The number of permutations without repetition is given

1. **Problem Statement:** Find the number of runs of 1 of length $m$ in a random binary string of length $n$ using generating functions. 2. **Understanding the problem:** A run of

1. **Problem statement:** Find the number of runs of length $m$ in a string of length $n$. 2. **Definition:** A run is a maximal substring of consecutive identical characters.

1. **Problem Statement:** Find the generating function for the number of runs of 1 in binary strings of size $n$. 2. **Understanding the Problem:** A run of 1s is a maximal consecu