📘 combinatorics

1. Problem a: Find the number of committees of 5 members from 8 women and 4 men with at most 2 men. - "At most 2 men" means 0, 1, or 2 men.

1. **Problem statement:** We need to find the number of paths a checker can take from the bottom-center position to the top of the board, moving only diagonally upward. The checker

1. The problem involves analyzing a grid with blocked squares marked by "X" and a red circle indicating a position on the board. 2. The grid has 8 columns and 6 rows, with blocked

1. **Define terms with examples:** 1. Permutation: An arrangement of objects in a specific order. Example: Arranging 3 books on a shelf in order (ABC, ACB, BAC, etc.).

1. **Define terms:** 1.a. Permutation: An arrangement of objects in a specific order. Example: Arranging 3 books on a shelf in order.

1. **Problem Statement:** We need to find the number of arrangements of the word "ABSOLUTE" (8 distinct letters) such that the word starts with a vowel (A, E, O, or U) and ends wit

1. **Difference between permutations and combinations:** Permutations consider the order of selection important, while combinations do not.

1. **Problem statement:** We want to find the number of nondecreasing sequences of natural numbers \(\langle a_1, a_2, \ldots, a_k \rangle\) with sum 49 that are *not* good. A sequ

1. **Problem statement:** Find the number of 4-digit numbers formed from digits 0,1,2,3,4,5,6,7 such that each number contains the digit 1 at least once. 2. **Total 4-digit numbers

1. **State the problem:** We need to find the number of 4-digit numbers formed from the digits 0, 1, 2, 3, 4, 5, 6, 7 such that each number contains the digit 1 at least once. 2. *

1. **Problem Statement:** Understand real-life scenarios involving permutations and combinations. 2. **Permutations Scenario:** Suppose you want to arrange 3 different books on a s

1. **Problem statement:** We have a bag with 4 red tokens, 3 white tokens, and $b$ blue tokens. We draw three tokens under different conditions (with replacement, without replaceme

1. **Stating the problem:** We want to find the possible combinations of gum balls, candies, and toffees in a container such that there is exactly 1 gum ball in the first container

1. Masalah: Kita memiliki 12 pemain takraw dengan kemampuan hampir sama, akan dibagi menjadi 3 regu (A, B, C). Setiap regu terdiri dari 3 pemain inti dan 1 pemain pengganti, total

1. The problem asks for the total number of 4-digit passwords where each digit can be from 0 to 9 inclusive. 2. Since each digit can be any of 10 possible values (0 through 9), and

1. The problem asks for the number of possible choices for each digit in a 4-digit password. 2. Each digit can be any number from 0 to 9 inclusive.

1. **State the problem:** We have 12 people and want to form a committee of 5. From this committee, we select a President and a Secretary (distinct roles). Two people, X and Y, ref

1. **State the problem:** We need to find the number of different possible arrangements (permutations) of the letters in the word "bookkeeper". 2. **Count the letters:** The word "

1. **Problem 14.1.1:** How many different 4-digit numbers can be formed using digits 1, 3, 4, 6, 7, 9 if each digit is used once only? Step 1: We have 6 digits and want to form 4-d

1. The problem is to simplify the expression $15c5$. 2. Assuming $15c5$ represents a combination, it means the number of ways to choose 5 items from 15, denoted as $\binom{15}{5}$.

1. State the problem and recall Exercise 18. From Exercise 18 we have the explicit formula for derangements.