📘 combinatorics

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

1. **Problem statement:** We need to find the number of possible committees of 5 people chosen from 6 men and 4 women under three different conditions. 2. **(a) Committee with 3 me

1. First, calculate $15C_{10}$ using the formula for combinations: $$15C_{10} = \frac{15 \times 14 \times 13 \times 12 \times 11 \times 10!}{10! \times 5 \times 4 \times 3 \times 2

1. The problem asks to calculate the binomial coefficient $\binom{9}{2}$, which means "9 choose 2". 2. The binomial coefficient formula is given by:

1. The problem is to understand Pascal's Rule for binomial coefficients. 2. Pascal's Rule states that for any integers $n$ and $k$ with $0<k<n$, the binomial coefficients satisfy:

1. **Problem Statement:** We have 6 characters to form a password from letters \(\{b,f,g,k,m\}\), numbers \(\{3,5,7,9\}\), and symbols \(\{*,!,@\}\), with each character used at mo

1. **State the problem:** We need to find the number of 6-character passwords formed from letters {b, f, g, k, m}, numbers {3, 5, 7, 9}, and symbols {*, !, @} with given restrictio

1. The problem asks to find the number of interesting ordered quadruples $(p, q, r, s)$ of integers such that $1 \leq p < q < r < s \leq 10$. 2. Since $p, q, r, s$ are strictly inc

1. **Problem statement:** Show that $\binom{n}{r} + \binom{n}{r - 1} = \binom{n + 1}{r}$ and prove by induction the binomial theorem expansion:

1. **Problem statement:** A robotics team of 6 students is selected from 4 boys and 3 girls.