In this question, write down your answer, no need for any justification. You can leave your answer in terms of factorials, combination symbols, permutation symbols, etc. Please clearly box your answers in your sub- mission to Gradescope. (a) {1,2, 3, 4, 5, 6, 7}? How many injective functions are there from {1,2,3} to (b) the range of an injective function with domain {1, 2, 3}? How many subsets are there of {1,2,3, 4, 5, 6, 7} that are (c) group in order to guarantee that at least three people in the group were born in the same month (of possibly different years)? (The answer is not three; if you have three people they could all be born in the same month, but this is not guaranteed). What is the smallest number of people that must be in a (d) What is the coefficient of a b? in (a + b)"? (e) What is the coefficient of a bc in (a +b+ c)8?
Since you have asked a question with multiple subparts, we will solve first three subparts for you. To get remaining sub part solved please repost the complete question and mentioned the sub parts to be solved.
(a)
Let be an injective function. Therefore there are total choices for . Since the function is injective, therefore we have left with total choices for . Since function is injective, therefore we have left with total choices for .
Hence by the fundamental principle of counting,
Total possible injective function from to are .
(b)
We know that any injective function from a finite set to its range is always a bijective function, because each element will map to distinct elements and therefore the cardinality of range set will be equal to cardinality of the domain.
Now we need to find the number of subset of , that are the range of injective function with domain .
That means the range set will consist of elements. Hence we need to find total number of subset of having cardinality . Therefore we need to choose elements from the set of elements. Which can be done in ways.
Hence total such subset will be .
Trending now
This is a popular solution!
Step by step
Solved in 2 steps
