Description

1 Count it
Let’s get some practice with counting!
(a) How many sequences of 15 coin-flips are there that contain exactly 4 heads?
(b) An anagram of HALLOWEEN is any re-ordering of the letters of HALLOWEEN, i.e., any string made up of the letters H, A, L, L, O, W, E, E, N in any order. The anagram does not have to be an English word.
How many different anagrams of HALLOWEEN are there?
(c) How many solutions does y0 +y1 +···+yk =n have, if each y must be a non-negative integer?
(d) How many solutions does y0 +y1 =n have, if each y must be a positive integer?
(e) How many solutions does y0 +y1 +···+yk =n have, if each y must be a positive integer?
2 Inclusion and exclusion
What is total number of positive numbers that smaller than 100 and coprime to 100?
3 Identities

(Note: Assuming r > n,s > n)
4 Largest binom
For which value(s) of k is nk maximum? Prove your answer.