📘 combinatorics

1. **Stating the problem:** We have a symphony orchestra with 30 Haydn symphonies, 15 modern works, and 9 Beethoven symphonies. The program consists of one Haydn symphony, followed

1. **Problem statement:** We have 12 members (6 married couples) in a committee.

1. **Problem statement:** There are 10 members sitting around a round table, including 1 chairman seat. Among them, 4 members form a written report subcommittee. We want to find th

1. **State the problem:** A spinner with eight sections labeled A through H is spun twice. We need to find the total number of possible outcomes. 2. **Understand the problem:** Eac

1. **State the problem:** We want to find how many 5-letter arrangements can be made from the 26 letters of the English alphabet with no repeated letters. 2. **Formula used:** The

1. **Énoncé du problème :** Montrer l'identité de Vandermonde suivante :

1. The problem is to find the number of combinations of 3 items chosen from a set of $n$ items. 2. The formula for combinations is given by:

1. **Problem statement:** Calculate the number of different serial numbers possible on a dollar bill where the serial number consists of a letter, followed by eight digits, and the

1. **State the problem:** We have a class of 30 kids and want to select 4 of them to write their exam in the library. We need to find how many ways this selection can be done. 2. *

1. Problem: Find the number of different three-digit numbers under various conditions. 2. Formula and rules: For counting numbers with digits, use permutations and combinations.

1. The problem asks to prove a statement involving nonnegative integers $n$ and $r$. 2. Since the exact statement to prove is missing, let's assume it involves a common combinatori

1. **Problem:** Find the number of ways to seat 8 choristers around a round table such that two particular choristers must sit together. 2. **Formula and Explanation:** When two pa

1. **Problem statement:** A student must answer 10 out of 13 questions. (i) He must answer at least the first two from the first 5 questions.

1. The problem is to find the value of $C(8,5)$, which represents the number of combinations of 8 items taken 5 at a time. 2. The formula for combinations is:

1. **Problem statement:** We have an alphabet $\Omega = \{X, Y, Z, T, 0, 1, 2, 3, 4, 5, 6, 7\}$ with 12 characters, where 4 are letters ($X,Y,Z,T$) and 8 are digits ($0$ to $7$). W

1. **Problem statement:** We have an alphabet $\Omega = \{X, Y, Z, T, 0, 1, 2, 3, 4, 5, 6, 7\}$ with 12 characters: 4 letters ($X,Y,Z,T$) and 8 digits ($0$ to $7$). A Q-key of leng

1. **State the problem:** We need to find how many different committees of 3 men and 4 women can be formed from 8 men and 6 women. 2. **Formula used:** The number of ways to choose

1. **State the problem:** We want to find the number of ways to select 6 questions out of 10. 2. **Formula used:** The number of ways to choose $k$ items from $n$ items without reg

1. **Stating the problem:** Given the equation $P(n, 2) = C(n + 1, 3)$, find the value of $n$ that satisfies this.

1. **Stating the problem:** Given the equation $P(n, 2) = C(n + 1, 3)$, find the value of $n$ that satisfies this. 2. **Recall formulas:**

1. **State the problem:** There are 10 contestants in a speech contest, and we want to find how many possible ways the first, second, and third presenters can be chosen from these