📘 combinatorics

1. **State the problem:** Find the number of distinguishable permutations of the letters in the string "AAABBBCD". 2. **Formula used:** The number of distinguishable permutations o

1. The problem is to find the value of the binomial coefficient $\binom{9}{8}$. 2. The formula for a binomial coefficient is:

1. The problem asks for the value of the binomial coefficient $\binom{8}{9}$. 2. The binomial coefficient $\binom{n}{k}$ is defined as the number of ways to choose $k$ elements fro

1. **Problem:** A license plate must have 2 letters (not I or O) followed by 3 digits. The last digit cannot be zero. How many different plates can be made? 2. **Step 1: Determine

1. **State the problem:** We want to find the total number of possible ice cream combinations given 6 flavors, 4 toppings, and 2 types of cones.

1. **Problem statement:** We start with an unpainted wooden cube. We paint exactly three faces: one red, one blue, and one green. The other three faces remain unpainted. We want to

1. **State the problem:** We want to find all possible outcomes of answering a quiz with five true/false questions. 2. **Understanding the problem:** Each question has 2 possible a

1. The problem is to select 3 random football matches for each member: gb, jkb, Scouse, spurslad from the given fixtures list. 2. The fixtures are divided into leagues: Premier Lea

1. The problem is to select 3 random football matches for each member: gb, jkb, Scouse, and spurslad from the given list of fixtures. 2. Since the problem is about random selection

1. **Problem statement:** We have 6 points on a circle and want to find the number of polygons with 3 or 4 sides that can be formed by connecting these points. 2. **Formula and exp

1. **State the problem:** We need to find how many teams of 5 people can be created from 45 people registered for a basketball tournament. 2. **Formula used:** The number of teams

1. **Problem statement:** Calculate the binomial coefficient $\binom{8}{2}$. 2. **Formula:** The binomial coefficient $\binom{n}{k}$ is calculated by the formula:

1. **Problem statement:** We want to find how many different codes can be created with 2 letters, 2 digits, and 2 letters in that order, with no repeated letters or digits, and exa

1. **Problem statement:** (i) Find the number of different seating arrangements of eight friends sitting together in a row.

1. Das Problem besteht darin, alle möglichen Wurfkombinationen beim Dart zu finden, die eine bestimmte Punktzahl ergeben, beginnend mit 3 und dann 4. 2. Wir betrachten die mögliche

1. **Stating the problem:** We need to form a 5-digit number using digits from 0 to 9 without repetition. The number cannot start with zero and must be even.

1. The problem involves understanding factorials and combinations as given by the expressions $S! = 120$, $\frac{12!}{2!2!2!2!} = 28337600$, and $C_5^3 = 10$. 2. First, recognize t

1. **State the problem:** We want to find the number of ways to make 80 cents using only quarters (25 cents), dimes (10 cents), and nickels (5 cents). 2. **Set variables:** Let $q$

1. The problem asks: How many of the same color are there? This question is ambiguous without additional context such as the total number of items or colors involved. 2. To solve p

1. **Problem Statement:** We want to find the number of ways to arrange 8 objects into 3 places. This is a permutation problem because the order matters. 2. **Formula Used:** The n

1. **Problem statement:** Three runners compete in a race. We want to find the number of ways they can finish the race under two conditions: a) No tied places.