📘 combinatorics

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

1. The problem is to understand and solve questions related to permutations and combinations. 2. Permutations refer to the arrangement of objects where order matters, and combinati

1. **Problem:** Given $n=6$ and $r=3$, determine which expressions about permutations $P(6,3)$ and combinations $C(6,3)$ are true. 2. **Formulas:**

1. The problem involves calculating permutations and combinations for given values $n=6$ and $r=3$. 2. Recall the formulas:

1. **Problem statement:** We need to find the number of ways to arrange 5 boys and 6 girls in a line such that all 5 boys occupy 5 consecutive positions. 2. **Understanding the pro

1. Problem 17: Arrange 6 distinct people into 2 distinct rows with all people in one of the rows. Formula: Number of ways to assign each person to one of 2 rows = $2^6$.

1. **Problem Statement:** A photographer wants to arrange 6 distinct people into 2 distinct rows for a photo. How many ways can this be done if all 6 people must be in one of the t

1. **Problem Statement:** A photographer wants to arrange 6 distinct people into 2 distinct rows for a photo. How many ways can this be done if all 6 people must be in one of the t

1. **Stating the problem:** Find the value of $\binom{n}{0}$ for each positive integer $n$. 2. **Formula and explanation:** The binomial coefficient $\binom{n}{k}$ is defined as:

1. **Problem statement:** We want to find the number of ways to arrange 8 students in a line such that two specific students, a and b, are not standing next to each other. 2. **Tot

1. **State the problem:** There are 15 contestants in a race, and medals are awarded for gold, silver, and bronze. We need to find how many possible outcomes of medal winners there