📘 combinatorics

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.

1. **Stating the problem:** We have 9 sticks arranged to form a frame subdividing an equilateral triangle into 9 smaller triangles. 2. **Information given:** There are 3 balloons t

1. The problem asks to evaluate the expression $\sqrt{16c_6}$.\n\n2. First, let's interpret the notation. If $16c_6$ means the binomial coefficient $\binom{16}{6}$, then we calcula

1. **Problemstellung:** Ein Computer soll alle unterschiedlichen Anordnungen der 26 Buchstaben des Alphabets abspeichern. Es soll berechnet werden, wie lange dieser Vorgang dauert,

1. The problem is to understand whether a given selection or arrangement is a permutation or a combination. 2. Permutations refer to arrangements where order matters. For example,

1. **State the problem:** We want to find the number of paths from point X (bottom left black square) to point Y (top row, 4th black square from left) on an 8x8 checkerboard. We ha

1. **Problem statement:** Vasya forms 5-letter words using the letters A, B, C, D, E. These words must have exactly one A and exactly two Bs.

1. The problem is to find the number of different character sequences of length five to six that can be formed from the four-letter alphabet {A, C, G, T}. 2. The alphabet size is 4

1. **Stating the problem:** We want to find the number of words formed using the letters of the word DEPARTMENT with each letter used at most once. 2. The word DEPARTMENT has 10 le

1. **Restate the problem:** We have three companies C1, C2, and C3 with members 4, 5, and 6 respectively.