= 3!5! = 56 Strings. Example: How many bit strings of length 8 have at MOST three zeros? Answer: Since there are AT MOST 3 zeros, there may be zero, 1, 2, or 3 zeros. Example: How many bit strings of length 8 have at MOST three zeros? Example: How many bit strings of length 10 have the same...

This is the same as a bit string of length 6 with exactly two ones (the bars). There are C(6,2) = 15. Therefore, there are 15 ways of choosing the cookies. A more complicated example: Suppose that a cookie shop has 3 different kinds of cookies. How many different ways can 10 cookies be chosen?

Each time we take a bill, we put a star in the bin. Corresponds to taking $1, $5, $20, $100, $100. How to set up the problem: This time we have 6 bars separating the bins, and since we are taking 5 bills, there will be 5 stars. How many ways are there to arrange 6 bars and 5 stars?

Permutations of Indistinguishable Objects How many words can be made by reordering the letters SUCCESS? Answer There are three S"s, two C"s, and a single U and E. So the answer is NOT going to be 7!. Answer How many ways are there to put the 3 S"s?

How many ways are there to distribute hands of 5 cards to each of four players from the standard deck of 52 cards?

For each exercise, use one of the planning grids provided in this handout to sketch the desired graphic. For each sketch, provide a full inventory of the graphical components and their attributes. For example, if a circle appears, we need to know the following details: Where is the bounding...

The lines tangent to the curve y = 2x at x = 1, at x = 2, and at x = 8. 8. Let F be the function given by F (x) = f (x)g(x), where f and g are functions for which f (x) and g (x) are both dened for all real values of x. (a) What is F (x)?

Do the following problems 1. Let f denote a real valued function defined on some open interval around a R. Consider a line of slope m and equation L(x) = f (a) + m(x - a) for all x R. Suppose that this line if the best approximation to the function f at a in the sense that |E(x)| lim = 0, xa |x...

2. Suppose that A is a set with n (A) = 12. (a) How many dierent subsets does A have? 212 = 4096 (b) How many dierent 11 element subsets does A have? 12 12 (c) How many dierent 2 element subsets does A have? 3. Suppose that S and T are sets such that n (S) = 21, n (T ) = 17, and n (S [ T ) =...

13. If an arrow is shot upward on the moon with a velocity of 58 m/s, its height(in meters) after t seconds is given by h(t) = 58t - 0.83t2 . (a) Find the velocity of the arrow after 1 s. (b) Find the velocity of the arrow when t = a. (c) When will the arrow hit the moon? (d) With what velocity...