📘 combinatorics

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

1. **Problem statement:** We have a committee of 6 people sitting in a row. Two specific members refuse to sit next to each other. We need to find the number of possible seating or

1. **Problem Statement:** Find the number of partitions of $n=7$ into (i) odd summands and (ii) even summands using generating functions. Then verify by listing all partitions. 2.

1. **Problem Statement:** Find the number of partitions of $n=7$ into (i) odd summands and (ii) even summands using generating functions. Then verify by listing all partitions. 2.

1. **Problem Statement:** Alice has a safe secured by a 4-digit code with no repeated digits. We want to find the number of possible codes that satisfy this condition. 2. **Formula

1. The problem is to find the total number of rugby matches played if each team plays every other team exactly once. 2. The formula to calculate the number of matches in a round-ro

1. **Problem statement:** Helen wants to choose one main course and one dessert from 6 main courses and 8 desserts. 2. **Formula for combinations:** When choosing one item from eac

1. **Problem Statement:** We want to find for how many positive integers $n$ from 1 to 2008, the set $\{1,2,3,\ldots,4n\}$ can be partitioned into $n$ disjoint subsets each contain

1. 問題陳述： 計算一個由4行5列小長方形組成的矩形網格中，總共有多少個長方形。

1. **Stating the problem:** Calculate the permutation $18P17$, which represents the number of ways to arrange 17 objects out of 18 distinct objects. 2. **Formula used:** The permut

1. **Problem statement:** We have 2 different meat tortillas and 4 different vegetable tortillas to arrange on a plate. We want to find the number of ways to arrange them under dif

1. **Stating the problem:** We want to arrange 4 textbooks, 3 exercise books, and 2 manuals on a shelf. We need to find the number of ways to do this under different conditions:

1. **Problem statement:** We have 10 athletes standing in a row for a photo, and the tallest athlete must be positioned exactly in the center. 2. **Understanding the problem:** Sin