📘 combinatorics

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

1. **State the problem:** Calculate the combination with repetition for $n=28$ and $r=13$. 2. **Formula:** The formula for combinations with repetition is $$\binom{n+r-1}{r} = \fra

1. **State the problem:** Calculate the combination formula given by $$\frac{(n+r-1)!}{r!(n-1)!}$$ for the values $n=35$ and $r=10$.

1. Masalah: Diberikan angka 0, 1, 3, 4, 5, 7, dan 8. Tentukan banyaknya bilangan genap tiga angka berbeda yang dapat disusun dari angka-angka tersebut. 2. Bilangan genap berarti an

1. The problem asks for the number of permutations of the digits 5, 4, 3, and 2. 2. A permutation is an arrangement of all the members of a set into some sequence or order.

1. **Problem statement:** We need to find the number of ways a basketball coach can select his first 5 players from a 15-man basketball team. 2. **Formula used:** This is a combina

1. **Stating the problem:** We want to understand permutations, which are arrangements of objects in a specific order. 2. **Formula for permutations:** The number of ways to arrang

1. The problem asks: How many rectangles of all sizes are there in a 3x3 grid of squares? Remember, squares are also rectangles. 2. To find the total number of rectangles in a grid

1. Masalah: Tentukan banyak bilangan genap yang terdiri dari 3 angka berbeda yang tersusun dari angka 2, 3, 4, 6, 7, dan 9. 2. Aturan: Bilangan genap adalah bilangan yang digit ter

1. **State the problem:** We need to find how many different posters can be made using one poster board and one marker. 2. **Identify the given information:** There are 3 different