Question

A sonnet is a 14-line poem in which certain rhyming patterns are followed. The writer Raymond Queneau published a book containing just 10 sonnets, each on a different page. However, these were structured such that other sonnets could be created as follows: the first line of a sonnet could come from the first line on any of the 10 pages, the second line could come from the second line on any of the 10 pages, and so on (successive lines were perforated for this purpose).
(a) How many sonnets can be created from the 10 in the book?
(b) If one of the sonnets counted in part (a) is selected at random, what is the probability that none of its lines came from either the first or the last sonnet in the book? (Round your answer to three decimal places.)

Answer 1

Answer:

a) $10^{14}$

b) 0.044

Step-by-step explanation:

Part a)

A Sonet is a 14-line poem. Raymond Queneau published a book containing just 10 sonnets, each on different pages. This means, on each page the writer wrote a 14-line poem. We have to find how many sonnets can be created from the 10 sonnets in the book.

Since, the first line of the new sonnet can be the first line of any of the 10 sonnets. So, there are 10 ways to select the first line. Similarly, the second line of the new sonnet can be the second line of any of the 10 sonnets. So, there are 10 ways to select the second line. Same goes for all the 14 lines i.e. there are 10 ways to select each of the line.

According to the fundamental principle of counting, the total number of possible sonnets would be the product of all the possibilities of all 14 lines.

So,

The number of sonnets that can be created from the book = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = $10^{14}$ sonnets

So, $10^{14}$ can be created from the 10 in the book.

Part b)

First we need to find how many such sonnets can be created that have none of their lines from 1st and last page. Since, total number of pages is 10, if we are NOT to select from 1st and last page, this will leave us with 8 pages (8 sonnets).

So, now the number of possible options for each line of the sonnet would be 8. And according to the  fundamental principle of counting, the number of sonnets with neither the line from 1st nor the last page would be = $8^{14}$

This represents the number of favorable outcomes, as we want the randomly selected sonnet to be such that none of its lines came from either the first or the last sonnet in the book.

So,

Number of favorable outcomes = $8^{14}$

Total possible outcomes of the event = $10^{14}$

As the probability is defined as the ratio of favorable outcomes to the total outcomes, we can write:

The probability that none of its lines came from either the first or the last sonnet in the book = $\frac{8^{14}}{10^{14}} = 0.044$