Chapter 6: Q32E (page 406)
Show that if there are 100,000,000 wage earners in the United States who earn less than 1,000,000 dollars (but at least a penny), then there are two who earned exactly the same amount of money, to the penny, last year.
The resultant answer is at least 2 people who get the same wage.
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Find the value of each of these quantities.
a) \(C(5,1)\)
b) \(C(5,3)\)
c) \(C(8,4)\)
d) \(C(8,8)\)
e) \(C(8,0)\)
f) \(C(12,6)\)
This procedure is used to break ties in games in the championship round of the World Cup soccer tournament. Each team selects five players in a prescribed order. Each of these players takes a penalty kick, with a player from the first team followed by a player from the second team and so on, following the order of players specified. If the score is still tied at the end of the 10 penalty kicks, this procedure is repeated. If the score is still tied after 20 penalty kicks, a sudden-death shootout occurs, with the first team scoring an unanswered goal victorious.
a) How many different scoring scenarios are possible if the game is settled in the first round of 10 penalty kicks, where the round ends once it is impossible for a team to equal the number of goals scored by the other team?
b) How many different scoring scenarios for the first and second groups of penalty kicks are possible if the game is settled in the second round of 10 penalty kicks?
c) How many scoring scenarios are possible for the full set of penalty kicks if the game is settled with no more than 10 total additional kicks after the two rounds of five kicks for each team?
Prove the hockeystick identity\(\sum\limits_{k = 0}^r {\left( {\begin{array}{*{20}{c}}{n + k}\\k\end{array}} \right)} = \left( {\begin{array}{*{20}{c}}{n + r + 1}\\r\end{array}} \right)\).whenever\(n\)and\(r\)are positive integers,
a) using a combinatorial argument.
b) using Pascal's identity.
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.)
How many ways are there for eight men and five women to stand in a line so that no two women stand next to each other? (Hint: First position the men and then consider possible positions for the women.)
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