Binomial coefficients are a family of positive integers that have a number of useful properties and they can be defined in several ways. one way to define them is as an indexed recursive function, c(n, k), where the "c" stands for "choice" or "combinations." in this case, the definition is as follows: c(n, 0) = 1, c(n, n) = 1, and, for 0 < k < n, c(n, k) = c(n − 1, k − 1) + c(n − 1, k). (a) show that, if we don’t use memoization, and n is even, then the running time for computing c(n, n/2) is at least 2n/2. (b) describe a scheme for computing c(n, k) using memoization. give a bigoh characterization of the number of arithmetic operations needed for computing c(n, n/2) in this case.

Using the artofstat website, run a permutation test to see if there is a difference in the mean amount of time spent on emails per day between in state and out of state students. (make sure to generate 10,000 permutations.) in state students: 2 3 3 6 2 1 1 5 3 2.5 out of state students: 1 2 2 1 2 1 4 3 9 1 10 1 3 what is the p-value?

Requirement types discussed during software development include functional and color scheme nonfunctional and code style constraint and nonfunctional fashionable and functional.

Write a modular program (no classes yet, just from what you learned last year), that allows two players to play a game of tic-tac-toe. use a two-dimensional char array with 3 rows and 3 columns as the game board. each element of the array should be initialized with an asterisk (*). the program should display the initial board configuration and then start a loop that does the following: allow player 1 to select a location on the board for an x by entering a row and column number. then redisplay the board with an x replacing the * in the chosen location. if there is no winner yet and the board is not yet full, allow player 2 to select a location on the board for an o by entering a row and column number. then redisplay the board with an o replacing the * in the chosen location. the loop should continue until a player has won or a tie has occurred, then display a message indicating who won, or reporting that a tie occurred. player 1 wins when there are three xs in a row, a column, or a diagonal on the game board. player 2 wins when there are three ox in a row, a column, or a diagonal on the game board. a tie occurs when all of the locations on the board are full, but there is no winner. input validation: only allow legal moves to be entered. the row must be 1, 2, or 3. the column must be 1, 2 3. the (row, column) position entered must currently be empty (i.e., still have an asterisk in it).