📘 combinatorics

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

1. Problem: How many routes are there from the lake to the cabins if a hiker can take 4 trails to the lake and then 3 trails from the lake to the cabins? 2. Formula: The total numb

1. **Problem:** Hendro memiliki 5 baju, 6 celana, dan 3 sepatu. Berapa banyak cara Hendro memasangkan pakaian yang dimilikinya? 2. **Formula:** Untuk menghitung banyak cara memasan

1. **Problem statement:** There are 40 members in a club including Ranuf and Saed. 35 members will travel in a coach and 5 in a car. Ranuf must be in the coach and Saed must be in

1. **Problem statement:** We have 400 students with scores from 0 to 10. For part (a), we know for each $i \in \{1,2,3,4,5\}$, the number of students scoring $i+5$ is 2 more than t

1. The problem is to calculate the combination $C(3,10)$, which represents the number of ways to choose 3 items from 10 without regard to order. 2. The formula for combinations is:

1. The problem is to evaluate the expression involving combinations: $$8C_0 \cdot 12C_4 + 8C_1 \cdot 12C_3 + 8C_2 \cdot 12C_2$$ which represents selecting groups from two sets. 2.

1. **Problem statement:** You want to buy 5 movies in total: 3 sci-fi and 2 western. The store has 12 sci-fi movies and 8 western movies. How many ways can you choose these movies?

1. The problem is to find the number of ways to choose 3 items from a set of $n$ items using combinations. 2. The formula for combinations (choosing $k$ items from $n$ without rega

1. **Problem statement:** Twelve people (7 Canadians and 5 Australians) apply for 5 jobs at a ski resort. We want to find the number of ways to award these jobs under different con

1. **Problem statement:** How many different identification codes can be made if the code consists of 2 numbers followed by 5 letters, the code cannot begin with 0, cannot contain

1. **Problem statement:** At Santa’s workshop, each Elf's ID consists of 2 numbers followed by 5 letters. The code cannot begin with 0, cannot contain the letter O, and no repetiti