📘 combinatorics

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

1. **State the problem:** We want to find the number of different amino-acid sequences possible for an octapeptide containing four of one amino acid, two of another, and two of a t

1. **Problem statement:** We need to find the number of different amino-acid sequences possible for an octapeptide containing four of one amino acid, two of another, and two of a t

1. **State the problem:** We need to find the number of permutations of selecting four leadership positions: Majority Leader and Assistant Majority Leader from 90 Democrats, and Mi

1. **State the problem:** We need to find the number of permutations of four leadership positions: Majority Leader and Assistant Majority Leader from 90 Democrats, and Minority Lea

1. **State the problem:** We need to find the number of permutations of selecting four leadership positions: Majority Leader and Assistant Majority Leader from 90 Democrats, and Mi

1. **Problem:** There are 5 towns and we want to arrange their pictures in a row. How many ways can they be arranged? 2. **Formula:** The number of ways to arrange $n$ distinct obj

1. The problem is to calculate the binomial coefficient $\binom{6}{3}$, which represents the number of ways to choose 3 elements from a set of 6 elements. 2. The formula for the bi

1. The problem states that you can use the same number twice. This typically applies to problems involving combinations, permutations, or equations where repetition is allowed. 2.

1. The problem is to understand the binomial coefficient $\binom{n}{k}$, which represents the number of ways to choose $k$ elements from a set of $n$ elements without regard to ord