Question

while making up his schedule for spring semester, tom complains that he doesnt have very many choices of schedule because of the general education requirements he has to meet. his advisor tells tom that he has to take one course from each of english (6 choices), history (4 choices), math/stats (8 choices), computer science (7 choices), and general science (6 choices). does tom have a legitimate gripe?
(a) if every possible course is available at the time hes registering, how many possible schedules can he choose from (disregarding when the classes meet)?
tom can choose from \\( \square \\) possible schedules.
(b) in an unprecedented effort to make the general education requirements more accessible, the dean of toms college decides to double the number of acceptable courses in each of those five areas. what effect does this have on the number of possible schedules?
tom can choose from \\( \square \\) possible schedules when the number of acceptable courses are doubled. the number of possible schedules is increased by a factor of \\( \square \\).

Explanation:

Step1: Use the multiplication principle for part (a)

The multiplication principle states that if there are \(m_1\) ways to do one thing, \(m_2\) ways to do a second thing, \(\cdots\), \(m_n\) ways to do an \(n\)th thing, then the total number of ways to do all \(n\) things together is \(m_1\times m_2\times\cdots\times m_n\).
Here, \(m_1 = 6\) (English courses), \(m_2=4\) (History courses), \(m_3 = 8\) (Math/Stats courses), \(m_4=7\) (Computer Science courses), \(m_5 = 6\) (General Science courses).
The number of possible schedules \(N_1=6\times4\times8\times7\times6\)

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Step2: Use the multiplication principle for part (b)

When the number of courses in each area is doubled, the new number of courses: \(m_1'=2\times6 = 12\), \(m_2'=2\times4=8\), \(m_3'=2\times8 = 16\), \(m_4'=2\times7=14\), \(m_5'=2\times6 = 12\)
The new number of possible schedules \(N_2=12\times8\times16\times14\times12\)

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The factor by which the number of schedules is increased is \(\frac{N_2}{N_1}\)
Since \(N_2=(2\times6)\times(2\times4)\times(2\times8)\times(2\times7)\times(2\times6)=2^5\times(6\times4\times8\times7\times6)\) and \(N_1 = 6\times4\times8\times7\times6\), \(\frac{N_2}{N_1}=2^5=32\)

Answer:

(a) \(8064\)
(b) \(258048\), \(32\)