📘 combinatorics

1. **Problem statement:** We need to find the number of ways to arrange an eight-oared boat crew from 11 men. Among them, 3 can steer but cannot row, and the rest cannot steer. Add

1. **State the problem:** We have 7 men and 6 women, total 13 runners. We want to form teams of 8 runners and find the proportion of teams with more women than men. 2. **Total numb

1. **Problem statement:** We have a basket with 8 mangoes, 4 oranges, and 2 mangosteen. We want to find the number of ways to pick 4 fruits simultaneously.

1. **Problem statement:** We need to find the number of ways to arrange 4 reindeer — Lancer, Prancer, Gloopin, and Bloopin — in a single-file line such that Lancer and Prancer are

1. **State the problem:** We want to find the number of unique ways to arrange the letters in the word HATTER. 2. **Identify the letters and their frequencies:** The word HATTER ha

1. **Stating the problem:** A school has a total of 78 possible pairs of students. We need to find how many students are participating. 2. **Formula used:** The number of ways to f

1. **Problem statement:** Freddie has a 6-digit code with no repeated digits and the first digit is not zero.

1. **State the problem:** We want to find the number of distinguishable ways to arrange the letters in the word "statistics". 2. **Formula used:** The number of distinguishable per

1. **Problem Statement:** A relief agency has 12 distinct aid kits (4 Medical (M), 5 Food (F), 3 WASH (W)) to distribute to 5 distinct districts (A, B, C, D, E).

1. **Problem statement:** Find the number of ways to arrange the letters of "olympic" under two conditions: 2. **Part (a): No restrictions**

1. **Problem statement:** Amy and her 4 close friends stand in a line with Amy exactly in the middle. We need to find how many different ways they can stand in such a line. 2. **Un

1. **Problem statement:** We have socks of 5 colors with quantities: red 90, pink 80, green 70, black 60, blue 50.

1. **Problem statement:** We have a bag with 5 blue marbles, 3 red marbles, and 5 white marbles. We pick marbles one by one without replacement. We want to find the number of ways

1. Let's restate the problem correctly based on your clarification. We want to analyze the inequality involving the summation on the left side:

1. **State the problem:** We need to find how many whole numbers between 100 and 400 contain the digit 2. 2. **Understand the range:** The numbers are from 101 to 399 (since 100 an

1. **State the problem:** We want to find how many four-letter words can be formed from the letters of the word ATTENTIVENESS, and how many of these words have at least one vowel.

1. **State the problem:** We want to find how many different permutations can be made from the letters of the word "help". 2. **Formula used:** The number of permutations of $n$ di

1. The problem is to determine the value of $16C2$, which represents the number of combinations of 16 items taken 2 at a time. 2. The formula for combinations is:

1. **State the problem:** We need to find how many subsets of the set $S$, where $S$ is the set of factors of 2026, contain exactly 2 elements. 2. **Find the factors of 2026:** Fir

1. Soal pertama meminta banyak susunan pengurus ketua, sekretaris, dan bendahara dari 20 anggota. 2. Rumus permutasi untuk memilih dan menyusun $r$ orang dari $n$ adalah $$P(n,r) =

1. Soal: Banyak susunan berfoto berjajar untuk 3 pasang pemain bulutangkis ganda dengan tidak ada setiap pemain dan pasangannya berdekatan. Langkah 1: Misalkan setiap pasangan dian