Chapter 6: Q35E (page 406)
There are 38 different time periods during which classes at a university can be scheduled. If there are 677 different classes, how many different rooms will be needed?
The resultant answer is 18 classrooms.
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Find the expansion of
a) using combinatorial reasoning, as in Example
b) using the binomial theorem.
Find the coefficient of.
How many ways are there to choose 6 items from 10 distinct items when
a) the items in the choices are ordered and repetition is not allowed?
b) the items in the choices are ordered and repetition is allowed?
c) the items in the choices are unordered and repetition is not allowed?
d) the items in the choices are unordered and repetition is allowed?
Give a combinatorial proof that \(\sum\limits_{k = 1}^n k \left( {\begin{array}{*{20}{l}}n\\k\end{array}} \right) = n{2^{n - 1}}\). (Hint: Count in two ways the number of ways to select a committee and to then select a leader of the committee.)
Let\(n\)and \(k\) be integers with \(1 \le k \le n\). Show that
\(\sum\limits_{k = 1}^n {\left( {\begin{array}{*{20}{l}}n\\k\end{array}} \right)} \left( {\begin{array}{*{20}{c}}n\\{k - 1}\end{array}} \right) = \left( {\begin{array}{*{20}{c}}{2n + 2}\\{n + 1}\end{array}} \right)/2 - \left( {\begin{array}{*{20}{c}}{2n}\\n\end{array}} \right)\)
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