13. Consider functions f: {1,2,3,4} {1, 2, 3, 4, 5, 6}. → a. How many functions are there total? b. How many functions are injective? c. How many of the injective functions are increasing? To be increasing means that if a < b then f(a) f(b), or in other words, the outputs get larger as the inputs get larger. 14. We have seen that the formula for P(n, k) is n! (n-k)! Your task here is to explain why this is the right formula. a. Suppose you have 12 chips, each a different color. How many different stacks of 5 chips can you make? Explain your answer and why it is the same as using the formula for P(12,5). b. Using the scenario of the 12 chips again, what does 12! count? What does 7! count? Explain. c. Explain why it makes sense to divide 12! by 7! when computing P(12,5) (in terms of the chips). d. Does your explanation work for numbers other than 12 and 5? Explain the formula P(n, k) = using the variables n and k. n! (n-k)!
13. Consider functions f: {1,2,3,4} {1, 2, 3, 4, 5, 6}. → a. How many functions are there total? b. How many functions are injective? c. How many of the injective functions are increasing? To be increasing means that if a < b then f(a) f(b), or in other words, the outputs get larger as the inputs get larger. 14. We have seen that the formula for P(n, k) is n! (n-k)! Your task here is to explain why this is the right formula. a. Suppose you have 12 chips, each a different color. How many different stacks of 5 chips can you make? Explain your answer and why it is the same as using the formula for P(12,5). b. Using the scenario of the 12 chips again, what does 12! count? What does 7! count? Explain. c. Explain why it makes sense to divide 12! by 7! when computing P(12,5) (in terms of the chips). d. Does your explanation work for numbers other than 12 and 5? Explain the formula P(n, k) = using the variables n and k. n! (n-k)!
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Combinatorics. Show solution to your answer.
Transcribed Image Text:13. Consider functions f: {1,2,3,4} {1, 2, 3, 4, 5, 6}.
a. How many functions are there total?
b. How many functions are injective?
c. How many of the injective functions are increasing? To be increasing
means that if a < b then f(a) < f(b), or in other words, the outputs get
larger as the inputs get larger.
n!
14. We have seen that the formula for P(n, k) is
. Your task here is to
(n - k)!
explain why this is the right formula.
a. Suppose you have 12 chips, each a different color. How many different
stacks of 5 chips can you make? Explain your answer and why it is the
same as using the formula for P(12,5).
b. Using the scenario of the 12 chips again, what does 12! count? What does
7! count? Explain.
c. Explain why it makes sense to divide 12! by 7! when computing P(12,5) (in
terms of the chips).
d. Does your explanation work for numbers other than 12 and 5? Explain the
formula P(n, k)
n!
=
(k) using the variables n and k.
(n-k)!
Transcribed Image Text:7. An anagram of a word is just a rearrangement of its letters. How many
different anagrams of "uncopyrightable" are there? (This happens to be the
longest common English word without any repeated letters.)
8. How many anagrams are there of the word "assesses" that start with the
letter "a"?
Solution
9. How many anagrams are there of "anagram"?
10. On a business retreat, your company of 20 businessmen and
businesswomen go golfing.
a. You need to divide up into foursomes (groups of 4 people): a first
foursome, a second foursome, and so on. How many ways can you do
this?
b. After all your hard work, you realize that in fact, you want each foursome
to include one of the five Board members. How many ways can you do
this?
Solution
11. How many different seating arrangements are possible for King Arthur and
his 9 knights around their round table?
Solution
12. Consider sets A and B with |A| = 10 and |B| = 17.
a. How many functions f: A → B are there?
b. How many functions f: AB are injective?
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