📘 combinatorics

1. **Problem statement:** We need to form a group of 5 people from 3 married couples (6 people), 4 male staff, and 5 female staff, with the condition that no married couple is incl

1. **Problem statement:** Determine the number of pathways from point A to point B on a grid where you can only move right or down. 2. **Understanding the grid:** The grid has 3 ro

1. Statement of the problem. After each round of a knock-out netball competition the losing teams drop out until just the winner remains.

1. The problem is to find the number of ways to choose 10 items from a set of $n$ items, which is a combination problem. 2. The formula for combinations is given by:

1. The problem is to generate 60 choose questions, which typically involve combinations. 2. The formula for combinations is given by $$C(n, k) = \frac{n!}{k!(n-k)!}$$ where $n$ is

1. **Problem statement:** We need to find the number of ways to divide 17 chapters among four writers such that the first and third writers write 5 chapters each, the second writer

1. The problem asks to find a trio (group of three digits) from the number 123456 without repeating any pair of digits. 2. We interpret "without repeating a pair" as selecting thre

1. The problem is to create a trio (group of three numbers) from the digits 1, 2, 3, 4, 5, 6 without repeating any sequence. 2. We interpret "without repeating a sequence" as formi

1. **Problem:** How many different 4-digit even numbers can be formed from the digits 1, 3, 5, 6, 8, and 9 if no repetition of digits is allowed? 2. **Formula and rules:**

1. **Problem statement:** We need to find the number of ways to arrange an eight-oared boat crew from 11 men. Among them, 3 can steer but cannot row, and the rest cannot steer. Add

1. **State the problem:** We have 7 men and 6 women, total 13 runners. We want to form teams of 8 runners and find the proportion of teams with more women than men. 2. **Total numb

1. **Problem statement:** We have a basket with 8 mangoes, 4 oranges, and 2 mangosteen. We want to find the number of ways to pick 4 fruits simultaneously.

1. **Problem statement:** We need to find the number of ways to arrange 4 reindeer — Lancer, Prancer, Gloopin, and Bloopin — in a single-file line such that Lancer and Prancer are

1. **State the problem:** We want to find the number of unique ways to arrange the letters in the word HATTER. 2. **Identify the letters and their frequencies:** The word HATTER ha

1. **Stating the problem:** A school has a total of 78 possible pairs of students. We need to find how many students are participating. 2. **Formula used:** The number of ways to f

1. **Problem statement:** Freddie has a 6-digit code with no repeated digits and the first digit is not zero.

1. **State the problem:** We want to find the number of distinguishable ways to arrange the letters in the word "statistics". 2. **Formula used:** The number of distinguishable per

1. **Problem Statement:** A relief agency has 12 distinct aid kits (4 Medical (M), 5 Food (F), 3 WASH (W)) to distribute to 5 distinct districts (A, B, C, D, E).

1. **Problem statement:** Find the number of ways to arrange the letters of "olympic" under two conditions: 2. **Part (a): No restrictions**

1. **Problem statement:** Amy and her 4 close friends stand in a line with Amy exactly in the middle. We need to find how many different ways they can stand in such a line. 2. **Un

1. **Problem statement:** We have socks of 5 colors with quantities: red 90, pink 80, green 70, black 60, blue 50.