📘 combinatorics

1. **Problem statement:** We need to find the number of ways to seat 6 girls and 2 boys in a row such that the two boys are always seated together. 2. **Understanding the problem:*

1. **Problem Statement:** Find the number of permutations of $n=4$ objects taken $r=2$ at a time. 2. **Formula:** The permutation formula is given by

1. **Problem:** Students are classified by gender (male or female), status (regular or irregular), and field of specialization (mathematics, physics, business, or languages). Find

1. **Problem:** Students are classified by gender (male, female), status (regular, irregular), and field of specialization (mathematics, physics, business, languages). Find all pos

1. **Problem:** Students are classified by gender (male or female), status (regular or irregular), and field of specialization (mathematics, physics, business, or languages). Find

1. **Problem:** Find the number of possible ways 5 people (Kenshin, Dan, Justin, Kris, Miguel) can be seated in a circular arrangement. 2. **Formula:** For circular permutations, t

1. **Problem:** Find the number of possible seating arrangements for Kenshin, Dan, Justin, Kris, and Miguel around a circular table. 2. **Formula:** For $n$ people seated around a

1. The problem is to calculate the binomial coefficient $\binom{20}{18}$. 2. The binomial coefficient formula is:

1. **Problem Statement:** We need to prove the recurrence relation $$D_n = nD_{n-1} + (-1)^n$$ for $$n \geq 1$$, where $$D_n$$ is defined as in Exercise 18 (usually the number of d

1. **Problem statement:** Find the number of ways 3 boys and 2 girls can sit in a row under different conditions. 2. **General formula for permutations:** The number of ways to arr

1. **Problem statement:** We have 12 students in a class, but one student is only there for refreshment and does not participate in experiments. So, effectively, 11 students are av

1. **Problem statement:** We need to find the number of ways to select 1 boy and 2 girls from a class of 27 boys and 14 girls. 2. **Formula used:** The number of ways to choose $k$

1. **Problem statement:** Given seven points on a plane, no three of which are collinear, we want to find how many polygons can be drawn using these points as vertices. 2. **Key id

1. The problem is to calculate the combination $C(10,4)$, which represents the number of ways to choose 4 items from 10 without regard to order. 2. The formula for combinations is:

1. The problem is to count items without double-counting those already included in brackets. 2. When counting, items inside brackets are considered part of the total and should not

1. The problem states that the number of permutations of $n$ objects taken 2 at a time is 12, i.e., $_nP_2 = 12$. 2. The formula for permutations of $n$ objects taken $r$ at a time

1. **Problem (a):** Find the number of different arrangements of the 10 letters in "ZOOLOGICAL" where the three Os are together and the two Ls are not next to each other. 2. **Step

1. The problem is to evaluate the product of combinations: $\binom{6}{2} \times \binom{4}{2} \times \binom{2}{2}$. 2. Recall the formula for combinations: $$\binom{n}{k} = \frac{n!

1. **Problem statement:** We have balls numbered from 1 to 2020 in a box. We draw balls without replacement. We want to find the minimum number of balls drawn to guarantee that amo

1. **Problem Statement:** (i) Find the number of possible seating arrangements for a family of 10 on a plane with 11 seats.

1. **Problem statement:** We want to find the number of ways to arrange twelve different amino acids into a polypeptide chain of length five. 2. **Formula used:** Since the order m