📘 combinatorics

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

1. **State the problem:** Calculate the value of $\binom{400}{2} \times 500$. 2. **Recall the formula for combinations:**

1. The problem is to select three questions from each difficulty level: easy, medium, and hard. 2. This is a combinatorial selection problem where we choose 3 items from each categ

1. The problem asks if letters and numbers can repeat in a given context. 2. To answer this, we need to understand the concept of repetition in combinatorics or arrangements.

1. **Problem Statement:** We need to find how many unique license plates can be formed with the format of three numbers followed by three letters, where both numbers and letters ca

1. **Problem statement:** Find the number of 5-digit odd numbers that can be formed with no repeated digits. 2. **Key points:**

1. **Problem statement:** We have 7 shortlisted applicants and 3 different internship roles. We want to find how many different ways the roles can be assigned when the order of app

1. **State the problem:** We want to find how many CVC (consonant–vowel–consonant) syllables can be formed using 10 consonants and 4 vowels. 2. **Formula and explanation:** The num

1. **Stating the problem:** We want to find the total number of combinations possible from a set of items. 2. **Formula used:** The number of combinations of choosing $k$ items fro

1. **Problem Statement:** We need to find the number of different combinations of candies taken from a bowl containing 7 identical kiwi, 10 identical cherry, and 7 identical orange

1. **Problem statement:** We have a circle divided into 6 sectors, each painted either red, green, or blue, with exactly 2 sectors of each color. We want to find the number of ways

1. The problem asks: How many different colorings are possible? We need more context to solve this, such as the number of objects to color and the number of colors available. 2. Ty

1. **Problem Statement:** We have a circle divided into 6 sectors of different sizes. We want to paint 2 sectors red, 2 green, and 2 blue such that no two adjacent sectors share th