How many poker hands of 6 cards can be dealt from a deck of 54 different cards?

In a computing system, a password consists of 4 characters. Each character can be an uppercase English letter, or a lowercase English letter, or a decimal digit. There is at least one digit in a valid password. How many valid passwords can we create?

A saleswoman needs to visit 9 cities in Puerto Rico in a sequence.

She wants to first visit San Juan. 

How many possible sequences does she have?

Suppose a function f: N \to Z is defined as:

1. Basis step: f(0) = 6

2. Recursive step: f(i) = 2 \times f(i-1) + 4 for all i \geq 1

What is f(1)?

Suppose a function f: N \to Z is defined as:

1. Basis step: f(0) = 5

2. Recursive step: f(i) = 3 \times f(i-1) + 4 for all i \geq 1

What is f(2)?

Let f( n ) be the minimal number of steps to solve the Tower of Hanoi game with n disks. f( n ) can be defined recursively as follows:

• The basis step: f(1)=1

• The recursive step: f( n )=2f(n-1)+1, when n > 1

Use mathematical induction to prove that, for any positive integer n, f( n ) = 2^n-1.

Given the following algorithm, matching each statement to the correct sequence for complexity analysis.

procedure bin_exp_mod(b, n, m): b, n and m are positive integers

1    x = 1

2    p = b mod m

3    while (n != 0) do

4        if (n mod 2 == 1)

5            x = (x * p) mod m

6        end of if

7        p = (p * p) mod m

8        n = n div 2

9    end of while

a    return x

Prove that if n is odd then n^2 + 5 \equiv 2 (mod 4).

Finding the remainder: -3 mod 5

Finding the quotient: -53 div 14