1. Find a positive integer X such that

2. A bagel shop sells six different kinds of bagels. Suppose you choose 20 bagels at random. What is the probability that your choice contains at least two bagels of each kind?

3. Find an integer t such that 0 < t < 98 and t = 7 (mod 9) and t = 6 (mod 11).

Recall that the roots of a polynomial f (3) are the solutions to f (3) = 0.  For example, the roots of 32 _ 53 + 6 are 2．3.

4. For an integer ) > 1 define the polynomial

(i)  (3 points) Find the roots of f1 (3), f2 (3), f3 (3).

(ii  (2 points) Find the roots of fn (3).

(iii)  (5 points) Prove that your answer to (ii) is correct.

5.  Define a poset Ⅹ．< as follows. The set Ⅹ consists of the subsets of {1．2．← ← ←．15}.  For

y．u e Ⅹ we define y < u whenever y C u . Fix 3 e Ⅹ with |3 | = 10. Find the number of

maximal chains in the poset that contain 3 .

7.  Define a poset Ⅹ．< as follows. The set Ⅹ consists of the positive integers that divide 24 33 = 432. Vertices y．u e Ⅹ satisfy y < u whenever y divides u .

(i)  (5 points) Find an antichain partition of Ⅹ into exactly 8 antichains.

(ii)  (5 points) Prove that any antichain partition of Ⅹ must involve at least 8 antichains.

8. Define the set ; = {y3．y2．y1．y0 }. List out all the subsets of ; using the squashed order.

9. For a positive even integer ), find the number of permutations of {1．2．← ← ←．)} for which the inversion sequence is symmetric.  By definition, an inversion sequence (b1．b2．← ← ←．bn ) is symmetric whenever bi  = bn −i+1  for 1 < i < ) .