Chapter 6: Q44SE (page 440)
How many ways are there to seat six boys and eight girls in a row of chairs so that no two boys are seated next to each other?
The number of different ways to seat six boys and eight girls in a row of chairs such that no two boys are seated next to each other is \({\rm{20321280}}\)ways.
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How many permutations of the letters \(ABCDEFGH\) contain
a) the string \(ED\)?
b) the string \(CDE\)?
c) the strings \(BA\) and \(FGH\)?
d) the strings \(AB\;,\;DE\) and \(GH\)?
e) the strings \(CAB\) and \(BED\)?
f) the strings \(BCA\) and \(ABF\)?
The internal telephone numbers in the phone system on a campus consist of five digits, with the first digit not equal to zero. How many different numbers can be assigned in this system?
The English alphabet contains \(21\) consonants and five vowels. How many strings of six lowercase letters of the English alphabet contain
a) exactly one vowel?
b) exactly two vowels?
c) at least one vowel?
d) at least two vowels?
How many solutions are there to the equation x1 + x2 + x3 + x4 + x5 = 21,
where xi, i = 1, 2, 3, 4, 5, is a nonnegative integer such that
a) x1 ≥ 1?
b) xi ≥ 2 for i = 1, 2, 3, 4, 5?
c) 0 ≤ x1 ≤ 10?
d) 0 ≤ x1 ≤ 3, 1 ≤ x2 < 4, and x3 ≥ 15?
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.)
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