Embibe Experts Solutions for Chapter: Combinatorics, Exercise 1: IOQM - PRMO and RMO 2020

Ria has $4$ green marbles and $8$ red marbles. She arranges them in a circle randomly, if the probability that no two green marbles are adjacent is $\frac{p}{q}$ where the positive integers $p, q$ have no common factors other than $1,$ what is $p+q?$

Let $A=\{1,2,3,4,5,6,7,8\}, B=\{9,10,11,12,13,14,15,16\}$ and $C=\{17,18,19,20,21,22,23,24\}.$ Find the number of triples $(x,y,z)$ such that $x\in A, y\in B, z\in C$ and $x+y+z=36$.

Find the largest positive integer $N$ such that the number of integers in the set $\{1,2,3,\dots ,N\}$ which are divisible by $3$ is equal to the number of integers which are divisible by $5$ or $7$ (or both).

Consider a permutation $\left(a_1,a_2,a_3,a_4,a_5\right)$ of $\{1,2,3,4,5\}.$ We say the $5$ -tuple $\left(a_1,a_2,a_3,a_4,a_5\right)$ is flawless if for all $1\le i<j<k\le 5,$ the sequence $\left(a_i,a_j,a_k\right)$ is not an arithmetic progression (in that order). Find the number of flawless $5$-tuples.

Ari chooses $7$ balls at random from $n$ balls numbered $1$ to $n.$ If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair of consecutive numbers in the chosen balls, find $n.$

Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different.)

In how many ways can a pair of parallel diagonals of a regular polygon of $10$ sides be selected?

Let $N$ be the number of ways of distributing $8$ chocolates of different brands among $3$ children such that each child gets at least one chocolate, and no two children get the same number of chocolates. Find the sum of the digits of $N.$