Question

A teacher has 7 textbooks to arrange on a shelf.

a) In how many ways can the 7 textbooks be arranged on the shelf?
Answer

b) If the textbooks are arranged at random, what is the probability that the textbooks will be arranged in alphabetical order? Enter your answer as a decimal, rounded to five decimal places.

Answer 1

a) The 7 textbooks can be arranged in a shelf in 7! = 5040 ways. This can be said because the first book in the shelf can be chosen from 7 books in 7C1=7 ways. Now out of the remaining 6 books, one can be selected in 6C1=6 ways. Hence the number of ways the first two positions in the shelf can be filled is 7*6 ways. Going by this procedure we can see that the 7 textbooks can be arranged in the shelf in 7*6*5*4*3*2*1 = 7! = 5040 ways.

b) If the textbooks are arranged randomly the probability that the textbooks will be arranged in alphabetical order is

(1/5040) = 0.0001984 = 0.0002.

This is so because there is only one particular way, where the arrangement of books will be in alphabetical order. Now if the books are arranged randomly then the total number of ways the books can be arranged is 5040 and out of them there is only one favorable case. Hence,

P[Textbooks are arranged in alphabetical order]

= (Number of ways textbooks can be arranged in alphabetical order)/(Total number of ways books can be arranged)

= 1/5040

.