Combinatorics | Number Theory Answers

How many ways are there to arrange the letters in the word ALGORITHMS? Note, the letter arrangement does not have to spell a word in the dictionary, but the new word must contain all the letters and each letter can be used only once.

Hint: Start with shorter words (e.g ALG or ALGO) and see if a pattern emerges with the number of possible arrangements.

1. How many 4 digits numbers can we create from 3,4,5,6,7,0,9 if;
2. We need all numbers to be even numbers        [2]
3. We need all of them to be odd numbers        [2]
4. We need all of them to be divisible by 5        [3]
5. In how many different ways can we choose a committee that is comprised of two men and one woman from a group of 5 men and 4 women if;
6. We need a chosen committee member is selected only once (i.e. if a person occupies only one position in the committee)                [2]
7. We need to choose a chairperson, a treasurer and secretary [3]

In how many ways can we arrange the letters of the word EXCELLENT to form different 9 letter words, with or without meaning   

Determine whether -104 is a quadratic residue or non residue of the prime 997.

Suppose p=0.9 and n=4 ,C={0000,1010,1100}.Compute pC,v where

v=1100

If a > 1, and  a | b , then 3b+1  mod a  =

Show that among any group of five (not necessarily consecutive) integers, there

are two with the same remainder when divided by 4

How many committees of three(3) can be formed from eight(8) people?

Suppose p=0.9 and n=4 ,C={0000,1010,1100}.Compute pC,v where

v=1100

Three planets P1, P2, P3 orbit a star in a distant galaxy. The perihelion of a

planet is the point in the orbit of a planet that is closest to the star. The orbital periods

of P1, P2, P3 are 3, 5 and 11 years respectively. The most recent perihelia of these

planets were in the years shown in the table below. What is the next year in which all

three planets achieve perihelion simultaneously?

Prove that 3n+4n+5n is divisible by 12 whenever n is an odd positive integer.