📘 combinatorics

1. **Problem statement:** From the digits 1, 2, 2, 3, 4, 5, form 4-digit numbers where each digit is used at most once. Conditions:

1. **Problem:** There are 18 mathematics majors and 325 computer science majors. (a) In how many ways can two representatives be picked so that one is a mathematics major and the o

1. **Problem Statement:** Find the number of 4-digit numbers that do not include the digits 7 and 8, are divisible by 4, and have no repeated digits. 2. **Key Points:**

1. **Problem statement:** From a standard 52-card deck, how many cards must be chosen at random to guarantee having at least three cards of clubs? 2. **Understanding the problem:**

1. **Stating the problem:** We have a deck of 52 cards, and we want to find two things:

1. **Problem statement:** How many ways can 6 people be lined up to get on a bus? 2. **Formula:** The number of ways to arrange $n$ distinct people in a line is $n!$.

1. **Problem Statement:** Calculate the number of possible telephone numbers, dressing combinations, security alarm codes, and four-digit numbers based on given constraints. 2. **T

1. **Problem Statement:** Determine the number of compositions of 1s and 2s that sum up to 18. 2. **Understanding the Problem:** A composition of a number is a way of writing it as

1. **State the problem:** We need to find how many distinct labels are possible for auditorium chairs, where each label consists of one uppercase letter followed by an integer from

1. Masalah: Tentukan berapa banyak bilangan ganjil antara 500 dan 1000 dengan ketentuan (i) semua angkanya berbeda, (ii) boleh ada angka yang berulang. 2. Definisi bilangan ganjil:

1. Masalah: Tentukan berapa banyak bilangan ganjil antara 500 dan 1000 dengan ketentuan (i) semua angkanya berbeda, (ii) boleh ada angka yang berulang. 2. Definisi bilangan ganjil:

1. Тодорхойлолт: 3x3 хэмжээтэй торон дээр гурван өөр хэмжээтэй квадрат зурж болдог гэж өгөгдсөн. 2. Ерөнхий зарчим: n x n хэмжээтэй торон дээр зурж болох квадратуудын тоо нь бүх бо

1. **Problem statement:** We need to select a group of 5 people, but 3 places are already taken. We want to find in how many ways the other 2 places can be filled using permutation

1. Problem: Find how many natural numbers $\leq 1000$ are not multiples of 4, 5, or 6. 2. Use the Inclusion-Exclusion Principle:

1. Problem: Find how many integers $\leq 1000$ are not multiples of 4, 5, or 6. 2. Use the Inclusion-Exclusion Principle:

1. The problem is to find the number of integers from 1 to 1000 that are not divisible by 2, 3, or 5. 2. We use the principle of inclusion-exclusion to solve this. The formula for

1. Асуудлыг тодорхойлъё: $3\times3$, $4\times4$, $5\times5$ хэмжээтэй торон дээр ялгаатай квадратуудыг тооцох. 2. Формул: $n\times n$ торон дээрх квадратуудын нийт тоо нь $$\sum_{k

1. **Problem:** Find the number of arrangements of the letters of the given words. 2. **Formula:** The number of arrangements of $n$ letters where some letters repeat is given by

1. **Problem statement:** Rudolf has four digits that can form the number 2025. He wants to know how many different numbers greater than 2025 can be made using these digits. 2. **D

1. **Problem statement:** We have 20 crews and 3 classes: Warrior, Archer, Mage.

1. **Problem statement:** We need to find the number of ways to select a committee of 5 members from 7 women and 9 men such that at least one woman is on the committee. 2. **Formul