📘 combinatorics

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.

1. The problem asks how many 5-letter words can be formed using the letters A, B, C, D, E where letter A appears exactly 3 times. 2. Since the word length is 5, and A occurs exactl

1. The problem asks for the number of different character sequences of length three to four that can be formed from a four-letter alphabet \{A, C, G, T\}. 2. Each position in the s

1. The problem asks to evaluate permutation expressions and find the number of unique permutations of letters in given sets. 2. Recall that permutation $nP r = \frac{n!}{(n-r)!}$.

1. Problem: Determine if 7 different colors are adequate to generate 42 unique color codes each consisting of 3 colors with no repetition. Step 1: Calculate the number of 3-color c

1. Given 11 people lined up for a photo. 2. The ketua (chairperson) must be in the middle position (which is position 6).

1. Problem: In how many ways can 10 classroom keys be arranged in a circular key chain? Solution:

1. The problem asks us to prove the binomial coefficient identity: $$\binom{n}{r} = \binom{n-1}{r} + \binom{n-1}{r-1}$$

1. Problem statement: We have 17 scientists discussing three particular topics. 2. Given: Any two scientists only discuss one of the three topics with each other.

1. Énoncé du problème :\nTrois scientifiques discutent de trois sujets précis. Entre deux scientifiques, une seule discussion porte sur un sujet unique. Nous devons prouver qu'il e

1. Find $r$ given $12P_{r-1} : 13P_{r-2} = 3 : 4$. - Recall permutation formula: $nP_r = \frac{n!}{(n-r)!}$.

1. Given: $12P_{r-1} : 13P_{r-2} = 3 : 4$. Recall permutation formula $nP_r = \frac{n!}{(n-r)!}$. Write ratio as $$\frac{\frac{12!}{(12-(r-1))!}}{\frac{13!}{(13-(r-2))!}} = \frac{3

1. **State the problem:** We have a box with 6 red balls, 4 green balls, and 3 blue balls. We want to find the number of ways to select 3 balls such that each ball is a different c