Chapter 6: Q4E (page 413)
Let. S = {1,2,3,4,5}
a) List all the 3-permutations of S.
b) List all the 3 -combinations of S.
(a) The resultant answer is:
(b) The resultant answer is:
Over 30 million students worldwide already upgrade their learning with Vaia!
All the tools & learning materials you need for study success - in one app.
Get started for free
a) State the generalized pigeonhole principle.
b) Explain how the generalized pigeonhole principle can be used to show that among any 91 integers, there are at least ten that end with the same digit.
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?
Show that if \(n\)and\(k\)are positive integers, then\(\left( {\begin{array}{*{20}{c}}{n + 1}\\k\end{array}} \right) = (n + 1)\left( {\begin{array}{*{20}{c}}n\\{k - 1}\end{array}} \right)/k\). Use this identity to construct an inductive definition of the binomial coefficients.
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?
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.)
What do you think about this solution?
We value your feedback to improve our textbook solutions.
