1. pigeonhole problems

Thank you for helping me
( I need to know how to solve such problems also. please solve with explanation )

2. 2 more here....

3. This should be the answer for the 1st question.

No of pigeonholes : 26 alphabets
No of pigeons : 30 students
Ceiling function[ 30 / 26 ] = 2

This should be the answer for the second question

Let x be the number of balls the woman needs to select
Ceiling function [ x / 2 ] = 3
x is 5.

The number of balls she needs to select = No of red balls + 3 blue balls = 10 + 3 = 13

When attemping this kind of questions, you need to visualise what can be the pigeons and what can be the pigeonhole and hopefully it will help to solve the problem. Hope it helps with the rest of the questions.

4. hint for the third problem: let the pigeonholes be the remainders when the numbers are divided by d.

there are d remainders: 0,1,2,...,d - 1

5. for problem 18: trial and error seemed simple enough, since we only have 10 terms, so i never bothered thinking about how to apply the pigeonhole principle to this. we have to pick the subsequence in order of the original sequence. by trial and error, an increasing maximal sequence is 5,6,10,15,21.

just start by picking a number and going from left to right picking numbers that are larger than it, trying to get as much as possible

6. hint for problem 16:

the pigeonholes are the address numbers

the houses are the pigeons