📘 combinatorics

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

1. **Problem:** In how many ways can you arrange 5 Mathematics books, 4 Science books, and 3 English books on a shelf such that books of the same subject are kept together? 2. **Fo

1. **Problem:** You want to arrange 12 canned goods in a row: 3 identical meat loaf cans, 4 identical tomato sauce cans, 2 identical sardine cans, and 3 identical corned beef cans.

1. Problem 1: Arranging Books on a Shelf Suppose you have 5 different books and you want to arrange 3 of them on a shelf. How many different ways can you arrange these 3 books?

1. The problem is to create 500 sets of 12 random numbers each from the numbers 1 to 30, such that each number appears the same number of times with every other number, and no set

1. The problem is to create 500 sets, each containing 12 random numbers selected from a set of 30 numbers. 2. Each number must appear the same number of times across all sets.

1. The problem is to generate 500 sets, each containing 12 unique numbers from 1 to 30, such that each number appears equally often with every other number, and no set is repeated.

1. Problem statement. 1. Generate 500 distinct subsets of size 12 from the set $\{1,2,\dots,30\}$ so that every pair of distinct numbers appears together the same number of times.

1. **Problem statement:** We need to find how many different ways 3 students can be chosen from a group of 10 to sit in the front row. 2. **Formula used:** Since the order in which

1. **Problem statement:** Find the number of permutations of the letters A, B, C, D, E, F, G, H that contain: (a) the string "ED" as a block.

1. **Problem Statement:** Find the number of permutations of the letters of the name "shakunthala" and determine how many words come before "shakunthala" if all permutations of the

1. The problem is to understand and solve an "all ways" problem, which typically involves finding all possible ways to do something, often related to counting or combinatorics. 2.