Pigeonhole principle

I need help with the following review question, I am unsure of how to solve it. 1. Explain how the pigeonhole principle can be used to show that among any 11 integers, at least two must have the same last digit. thanks.

Representing Graphs and Graph Isomorphism

I need help solving the problem attached, I've tried to solve it but I'm just not getting the right answer, I must not be using the right procedure. thanks

There are 7 drawers in a tool box. There are 100 tools. What is the largest number of tools that must reside in the same drawer?

Can you please show me how to solve the question step by step. 1. There are 7 drawers in a tool box. There are 100 tools. What is the largest number of tools that must reside in the same drawer?

How many different license plates can be made if each license plate consists of three letters followed by three digits or four letters followed by two digits?

Could someone show me how to solve the following. 20.How many different license plates can be made if each license plate consists of three letters followed by three digits or four letters followed by two digits?

Recursive Algorithm

I need help solving the following. 23. Give a recursive algorithm for computing n * a using only addition, where n is a positive integer and a is a real number. (add a to itself n times).

Binary expansion

I have to do a problem similar to the one below but I don't have any examples on how to solve. Can you help me solve this one so that I can be able to solve the one I have to do. The binary expansion of an integer is 101001. Show how you compute the equivalent decimal value.

Proofs

could someone show me how to solve this. Prove that there are no solutions in positive integers to equation x^4 + y^4 = 100

Discrete Structures

Please see attached file. Thank you.

Discrete Math

I need help in solving the attached problem. Thanks!

Discrete Structures - How many strings of eight letters (from A, B, C, ... Z) are there

Please see attached file. thank you.