Answer 12 and 14 please
Continue randomly drawing letter-cards and number-cards until all are matched. Record your pairings in tabular form on a sheet of paper to hand in stapled behind this assignment. On your table, designate the following columns: (a) System State, the numbers, (b) Component, the letter you drew to match the system-state number, and (c) Available Number of States, the number of system states that are available for that letter (component) to be assigned at the time you drew that particular match. The first number in column C is 26, the number of system states available before the first letter is drawn to occupy the first system state.
Arrange the numbers in the third column of the table in descending numerical order (largest to smallest), and write out these numbers lengthwise here:
26, 25, 24, 23, 22, 21, 20, 19, 18, 17, 16, 15, 14, 13, 12, 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1. If you were to multiply this string of numbers together, you would have the total possible number of states that this alphabetical system could assume. What number do you get?
4.03291461127E26
You may remember from a past math class that this number is a factorial, written x!, where x is the highest number in the sequence, and expanded out in the form x * (x-1) * (x-2) * (x-3) * ... * 2 * 1.
Just to make sure that you did not make a calculating error in E-10, enter "26" in your calculator and hit the "X!" button or its equivalent, if there is one on your calculator. What number did you get?
26!
In general, then, the degeneracy (Ω) of a system in which the state assumed by one component influences by mutual exclusion a state that another component can assume is defined by:
Ω = x!
where x = the number of states in the system.
The highly ordered alphabetical arrangement (A, B, C, D, ... X, Y, Z) and the one you got on your random drawing are only two of 26! possible arrangements of the letters of the alphabet. Using the definition of degeneracy (Ω) from equation (3), what is the degeneracy of the English alphabet?
