Factorials

3 Key excerpts on "Factorials"

• 

  eBook - ePub

  Discrete Mathematics with Ducks

  • sarah-marie belcastro, Sarah-Marie Belcastro(Authors)
  • 2018(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  why are you shouting numbers at me? Oops. Let’s have a...

  Definition 1.2 The notation n ! is pronounced n factorial and means

  n · ( n - 1 ) · ( n - 2 ) · ⋯ · 3 · 2 · 1

  . (It would be confusing if we pronounced it “En!!" excitedly, tempting though that might be.) We define

  0 ! = 1

  because there is exactly one way to rearrange nothing.

  Let us make a quick comparison between permutations of a set and subsets of that same set.

  Example 1.5 Does a set with n elements have more subsets or more ways of being ordered? Check out the data:

  At first, it looks like the subsets are in the lead. But then suddenly at

  n = 4

  , the permutations overtake them and go dashing ahead! To see that this trend continues and that permutations will win the race for even moderately large n , notice that when increasing n by 1, the number of subsets is multiplied by 2, whereas the number of permutations is multiplied by

  n + 1

  .

  Check Yourself

  Try to do at least two of these problems.

  1. You receive a shipment of 36 legs for stools to go with the stock of mass-manufactured stool seats you already have. How many stools can you complete?
  2. Suppose we wanted to place all nine different numbers onto a Sudoku board without reusing rows or columns—how many ways would there be to do it?
  3. On the other hand, what if we wanted to place nine 4s onto a Sudoku board without reusing rows or columns? (Again, we will ignore the fact that on a Sudoku board, a player also cannot have two of the same number appear in the same
     3 × 3
     block.) How many different ways would there be to make that placement?
  4. How many orderings are there of a , b , c , d , e , f , g , h , i , j , k , l , m , n , o , p ?

   

  6.6 Try This! Play with Powers and Permutations

  Enjoy these problems; the first three are quite fundamental.

  1. Let us return to sugar-numbers for a moment.
     1. Pull a sugar-number out of the bag of n sugar-numbers. How many ways are there to do this?
     2. Now pull another sugar-number out of the bag and put it next to the first sugar-number. How many ways are there to do this?

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• 

  eBook - ePub

  Discrete Encounters

  • Craig Bauer(Author)
  • 2020(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  https://monoskop.org/images/2/21/Cajori_Florian_A_History_of_Mathematical_Notations_2_Vols.pdf .

  With computing power being so prevalent today, there’s no need to expand the table of Factorials for higher values. However, in the past such a feature was considered useful in books dealing with this material. In 1625, Marin Mersenne (1588–1648) gave a table for the values 1! through 50! in his La Verite des Sciences contre les Sceptiques ou Pyrrhoniens (The Truth of the Sciences against the Sceptics or Pyrrhonians).7 Later, in his 1648 book, Harmonicorum libri XII , Mersenne expanded his table “up to 64!, a ninety-digit number and the largest factorial ever calculated up to then, although his table contains a number of errors.”8

  The Factorials wouldn’t be worthy of inclusion here if the only problem they were suited for was to calculate the number of ways a group of people can arrange themselves in a grocery store checkout lane. The concept is valuable because it arises in a wide range of contexts. Mersenne, for example, used 5! = 120 to answer the question of how many five note songs can be made from the set of five distinct notes ut, re, mi, fa, sol. He went on to list them all.9 He did the same for six notes, explicitly listing 6! = 720 songs.10

  Although the work detailed here may seem merely tedious, Mersenne investigated other combinatorial problems and has been described as “by far the most important Renaissance author in the history of combinatorics before Gottfried Wilhelm Leibniz”11 Also, Mersenne’s explicit listing of hundreds of items raises an important issue. It’s one thing to calculate how many of something there can be, but it is quite another matter to develop an algorithm for generating all of the possibilities. The latter is a task that often confronts computer programmers.

  There’s a great deal of fascinating material connected with the concept of Factorials. The pages that follow alternate between examples of this and some simple problems in combinatorics.

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• 

  eBook - ePub

  Basic Gambling Mathematics

  The Numbers Behind the Neon, Second Edition

  • Mark Bollman(Author)
  • 2023(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  On a video slot machine, the screen displays simulated reels, and the computer can process winning and losing combinations on dozens, even hundreds, of paylines that criss-cross the screen connecting symbols. As with computerized reel slots, a generated random number determines the arrangement of symbols on the virtual reels.

  Example 2.1.2. The Carnival of Mystery video slot machine, manufactured by International Game Technology, uses a five-reel screen with three symbols displayed on each reel. The game can be played with any symbol on each reel combining with any symbol on each other reel to form a payline. A payline is built by choosing one displayed symbol from the three on each reel, and so the total number of paylines is

  3 5

  = 243

  . On a penny machine, all 243 lines can be played for one wager of 25¢.■

  A special case of the Fundamental Counting Principle arises when we consider the number of ways to arrange a set of n elements, with no repetition allowed, in different orders. The first element may be chosen in n ways, the second in

  n − 1

  , and so on, down to the last item, which may be chosen in only one way. The total number of orders for a set of n elements is thus

  n · ( n − 1 ) · ( n − 2 ) · … · 3 · 2 · 1

  . This number is given a special name, nfactorial.

  Definition 2.1.1. If n is a positive integer, the factorial of n, denoted

  n !

  , is the product of all of the positive integers up to and including n:

  n ! = 1 · 2 · 3 · … · ( n − 1 ) · n .

  0 ! = 1

  , by definition.

  It is an immediate consequence of the definition that

  n ! = n · ( n − 1 ) ! .

  Factorials get very big very fast.

  4 ! = 24

  , but then

  5 ! = 120

  and

  6 ! = 720

  . 10! is greater than 1 million, and 52!, which is the number of different ways to arrange a standard deck of cards, is approximately

  8.066 ×

  10 67

  . In the 1970s and 1980s, the largest factorial that could be computed on a standard scientific calculator was

  69 ! ≈ 1.711 ×

  10 98

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