📘 combinatorics

1. **Problem statement:** We have a bag with 5 blue marbles, 3 red marbles, and 5 white marbles. We pick marbles one by one without replacement. We want to find the number of ways

1. Let's restate the problem correctly based on your clarification. We want to analyze the inequality involving the summation on the left side:

1. **State the problem:** We need to find how many whole numbers between 100 and 400 contain the digit 2. 2. **Understand the range:** The numbers are from 101 to 399 (since 100 an

1. **State the problem:** We want to find how many four-letter words can be formed from the letters of the word ATTENTIVENESS, and how many of these words have at least one vowel.

1. **State the problem:** We want to find how many different permutations can be made from the letters of the word "help". 2. **Formula used:** The number of permutations of $n$ di

1. The problem is to determine the value of $16C2$, which represents the number of combinations of 16 items taken 2 at a time. 2. The formula for combinations is:

1. **State the problem:** We need to find how many subsets of the set $S$, where $S$ is the set of factors of 2026, contain exactly 2 elements. 2. **Find the factors of 2026:** Fir

1. Soal pertama meminta banyak susunan pengurus ketua, sekretaris, dan bendahara dari 20 anggota. 2. Rumus permutasi untuk memilih dan menyusun $r$ orang dari $n$ adalah $$P(n,r) =

1. Soal: Banyak susunan berfoto berjajar untuk 3 pasang pemain bulutangkis ganda dengan tidak ada setiap pemain dan pasangannya berdekatan. Langkah 1: Misalkan setiap pasangan dian

1. Soal pertama: Banyak bilangan 4 angka berbeda lebih dari 4000 dari angka 2, 3, 4, 5, 6, 7. - Syarat: angka pertama harus 4, 5, 6, atau 7 agar > 4000.

1. The problem asks: How many 4-digit numbers greater than 4000 can be formed using the digits 2, 3, 4, 5, 6, and 7, with all digits different? 2. To solve this, we use the countin

1. **State the problem:** A restaurant menu has 3 sandwiches, 4 drinks, and 5 desserts. We want to find how many unique meals are possible if a meal consists of one item from each

1. **State the problem:** Julius wants to know how many different meal combinations he can have if he chooses one dish from each course: appetizer, main course, and dessert. 2. **F

1. **State the problem:** Edwin wants to buy three vehicles: a car, a truck, and a motorcycle. Each vehicle has specific color options. 2. **List the options:**

1. **Problem:** Find the generating function for $a_r$, the number of integer solutions to $e_1 + e_2 = r$ with $e_1 \in \{0,3,4,8\}$ and $e_2 \in \{0,4,5,8\}$. **Step 1:** The gen

1. **Problem:** Determine a generating function for the sequence $a_r$ given by the number of integer solutions to the equation $$e_1 + e_2 = r$$ with $$e_1 \in \{0,3,4,8\}$$ and $

1. **Problem statement:** We want to find the number of distinct paths from point $(0,0)$ to point $(3,3)$ on a grid, where movement is allowed only to the right, upward, or diagon

1. **Problem Statement:** We need to find the number of different paths from the bottom-left corner to the top-right corner of a 6 × 6 grid, moving only right or up, without crossi

1. **State the problem:** We have 24 points placed around a circle, and we want to find how many line segments are drawn between every pair of points. 2. **Formula used:** The numb

1. **Problem:** How many handshakes occur if eight people each shake hands with every other person exactly once? 2. **Formula:** The number of handshakes among $n$ people is given

1. The problem states that a teacher has 5 different pets and 16 volunteers out of 75 students. 2. The teacher will select 5 volunteers to take 1 pet each.