📘 combinatorics

1. **Stating the problem:** We have 9 sticks arranged to form a frame subdividing an equilateral triangle into 9 smaller triangles. 2. **Information given:** There are 3 balloons t

1. The problem asks to evaluate the expression $\sqrt{16c_6}$.\n\n2. First, let's interpret the notation. If $16c_6$ means the binomial coefficient $\binom{16}{6}$, then we calcula

1. **Problemstellung:** Ein Computer soll alle unterschiedlichen Anordnungen der 26 Buchstaben des Alphabets abspeichern. Es soll berechnet werden, wie lange dieser Vorgang dauert,

1. The problem is to understand whether a given selection or arrangement is a permutation or a combination. 2. Permutations refer to arrangements where order matters. For example,

1. **State the problem:** We want to find the number of paths from point X (bottom left black square) to point Y (top row, 4th black square from left) on an 8x8 checkerboard. We ha

1. **Problem statement:** Vasya forms 5-letter words using the letters A, B, C, D, E. These words must have exactly one A and exactly two Bs.

1. The problem is to find the number of different character sequences of length five to six that can be formed from the four-letter alphabet {A, C, G, T}. 2. The alphabet size is 4

1. **Stating the problem:** We want to find the number of words formed using the letters of the word DEPARTMENT with each letter used at most once. 2. The word DEPARTMENT has 10 le

1. **Restate the problem:** We have three companies C1, C2, and C3 with members 4, 5, and 6 respectively.

1. **Problem Statement:** We need to find the number of paths from point $X$ (bottom-left corner) to point $Y$ (top-right corner) on a $6 \times 6$ grid, moving only East or North

1. **State the problem:** We want to find the number of ways to travel from point $X$ at the bottom-left corner to point $Y$ at the top-right corner of a $6 \times 6$ grid, moving

1. **Stating the problem:** We need to count the number of ways to travel from point $X$ (bottom-left corner) to point $Y$ (top-right corner) on a $6 \times 6$ grid by moving only

1. The problem asks for the number of ways to travel from point $X$ at coordinate $(0,0)$ to point $Y$ at $(7,5)$ by moving only East or North, without passing through points $A(2,

1. **Problem Statement:** We must find the number of ways to travel from point $X$ at $(1,1)$ to point $Y$ at $(6,5)$ on a grid by moving only East or North without passing through

1. The problem asks for the number of ways to travel from point $X$ (assumed at $(0,0)$) to point $Y$ (assumed at $(6,6)$) on a 6x6 grid, moving only East (right) or North (up). 2.

1. **Problem statement:** We have the letters R, I, T, A, N, G, L, E arranged evenly around a circle of 26 letters spaced evenly in alphabetical order. The path length between lett

1. **Problem Statement:** Find the shortest and longest distances between permutations of the letters in "ritangle" (8 distinct letters) considering all $8!$ arrangements. 2. Since

1. **State the problem:** We have 8 letters \(\{R, I, T, A, N, G, L, E\}\) arranged around a circle of 26 evenly spaced points labeled A to Z clockwise. 2. The "path length" betwee

1. The problem asks: How many different teams of 3 can be chosen from a squad of 8? 2. This is a combination problem because the order of choosing team members does not matter.

1. Pernyataan masalah: Kita diminta menentukan banyaknya password berbeda yang dapat dibentuk dari 6 karakter dengan simbol yang tersedia yaitu 2, 4, 6, 9, P, S, dan B. 2. Jumlah k

1. **State the problem:** Grace has 16 jellybeans: 8 red, 4 green, and 4 blue. We want to find the minimum number she must take out to be sure she has at least one jellybean of eac