📘 combinatorics

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

1. **فهم المشكلة:** سؤالنا هو إيجاد جميع السلاسل من الأعداد التي مجموع أرقامها يساوي 27، ثم تحويل كل رقم في السلسلة إلى مجموع أرقامه الفردية للحصول على مجموع جديد يساوي 18 (كما في