There are four questions for 100 total points.

Question 3(c) corrected 16 November.

When your answer to a problem is a particular number, you may use
exponents, falling exponents, factorials, and/or the "choose" notation
to indicate the number. However, if the answer is *less than 100*,
you must give the number in ordinary decimal notation for full credit.

**Question 1 (20):**

A sushi bar sells five kinds of rolls: Alaska, California, Eel, Salmon, and Tuna. Six customers enter the bar and order one roll each.

- (a,5) In how many ways can each customer pick a type of roll? (Here a typical selection might be "Customer 1 -- Tuna, Customer 2 -- Eel, Customer 3 -- Tuna, Customer 4 -- Alaska, Customer 5 -- Tuna, Customer 6 -- Eel".)
- (b,5) The order will be reported to the sushi chef as simply the number of rolls of each type, so that the example above would be "Three Tuna, two Eel, and an Alaskan". In how many ways could the six rolls be ordered in this way?
- (c,5) How many possibilities are there for the
*set*of roll types ordered by the six customers? - (d,5) In how many ways could the six customers order without two or more of them ordering the same type of roll?

**Question 2 (30):**
Let n and k be arbitrary positive naturals with k ≤ n.

- (a,5) How many binary strings of length n have k or fewer ones?
- (b,5) How many binary strings of length n+1 have exactly k ones?
- (c,10) Prove that for any naturals n and k, the number of binary strings of length n+1 with k ones equals the number of binary strings of length n with either k or k-1 ones.
- (d,10) Prove
*by induction*that for all naturals n and k with k ≤ n, the number of binary strings of length n with k or fewer ones is greater than or equal to the number of binary strings of length n+1 with exactly k ones. (Hint: Let n be arbitary and use induction on k, starting with k=1.)

**Question 3 (30):**
If n and k are any naturals, define h(n,k) to be the number of subsets of
the set {1,...,n} of size k that do not contain two *adjacent* numbers.
For example, h(4,2) = 3 because the relevant sets are {1,3}, {1,4}, and {2,4}.
The other three subsets of size 2 contain two adjacent numbers.

- (a,10) Explain why h(n,0) = 1, h(n,1) = n, and h(n,2) = (n-1 choose 2).
- (b,5) Explain why h(n,k) = 0 if n is even and k > n/2.
- (c,15) Prove that h(n,k) = (n choose 2k-1) for all n and k. Sorry! The correct number is (n-k+1 choose k).. (Hint: Define and justify a bijection from the no-adjacent element sets of size k from {1,...,n} and all the sets of size k from {1,...,n-k+1}. You can also prove this by induction.)

**Question 4 (20)**:
The word "CAMBRIDGE" consists of nine different letters.

- (a,5) How many three-letter strings can be formed from the letters of "CAMBRIDGE" without repeating a letter?
- (b,5) If I choose one of the three-letter strings at random, what is the probability that I get one whose letters are in alphabetical order, as in the string "AIM"?
- (c,10) If I divide the nine letters into three three-letter strings with every division and every order of the strings being equally likely, what is the expected number of strings that will have their letters in alphabetical order? Justify your answer.

Last modified 16 November 2004