Combinatorics. Show solution to your answer. 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 increasingmeans that if a < b then f(a) < f(b), or in other words, the outputs getlarger 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 differentstacks of 5 chips can you make? Explain your answer and why it is thesame as using the formula for P(12,5).b. Using the scenario of the 12 chips again, what does 12! count? What does7! count? Explain.c. Explain why it makes sense to divide 12! by 7! when computing P(12,5) (interms of the chips).d. Does your explanation work for numbers other than 12 and 5? Explain theformula P(n, k)n!=(k) using the variables n and k.(n-k)! 7. An anagram of a word is just a rearrangement of its letters. How manydifferent anagrams of "uncopyrightable" are there? (This happens to be thelongest common English word without any repeated letters.)8. How many anagrams are there of the word "assesses" that start with theletter "a"?Solution9. How many anagrams are there of "anagram"?10. On a business retreat, your company of 20 businessmen andbusinesswomen go golfing.a. You need to divide up into foursomes (groups of 4 people): a firstfoursome, a second foursome, and so on. How many ways can you dothis?b. After all your hard work, you realize that in fact, you want each foursometo include one of the five Board members. How many ways can you dothis?Solution11. How many different seating arrangements are possible for King Arthur andhis 9 knights around their round table?Solution12. 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?

Since you have asked multiple questions , we will solve the first question for you. If you want…

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)!