# First Midterm Exam

### Directions:

• Answer the problems on the exam pages.
• There are six problems for 100 total points. Actual scale is A=90, C=60.
• If you need extra space use the back of a page.
• No books, notes, calculators, or collaboration.
• The first four questions are true/false, with five points for the correct boolean answer and up to five for a correct justification.
• The parts of Question 5 have numerical answers -- you may give your answer in the form of an expression using arithmetic operations, powers, falling powers, or the factorial function. If you give your answer using the "choose" notation, also give it using only operations on this list. In addition, if your answer is a non-negative integer less than or equal to 100, you must compute the number for full credit.

```  Q1: 10 points
Q2: 10 points
Q3: 10 points
Q4: 10 points
Q5: 30 points
Q6: 30 points
Total: 100 points
```

• The first two true-false questions involve a situation where three six-sided dice are thrown. A "throw" is an outcome such as "die 1 is 5, die 2 is 3, die 3 is 6".

• Question 1 (10): True or false with justification: The number of possible throws with at least one six is greater than the number of throws without any sixes.

• Question 2 (10): True or false with justification: The number of throws that contain at least one five and at least one six is exactly 63 - [53 + 53 - 43].

• Question 3 (10): True or false with justification: Assume that Zane's Noodle Stand offers 13 kinds of vegetables that may be put in your soup. Then the number of ways to choose exactly five different kinds of vegetables is strictly less than (13 choose 5).

• Question 4 (10): True or false with justification: Assume that eight horses are running a race, and that an "outcome" of the race is defined to be which horse is first, which is second, and which is third. Then there are over 300 possible outcomes of the race.

• Question 5 (30): A sushi bar offers four kinds of rolls: Alaskan, California, Salmon, and Tuna. Five customers come into the bar and order one roll each. The waiter writes down which customer ordered which kind of roll, so a "waiter-order" might be "1-A, 2-T, 3-T, 4-C, 5-A". The waiter then gives the sushi chef a "chef-order", which in this case would be "two A, one C, two T".
• (a,5) How many different possible waiter-orders are there from the five customers?
• (b,5) How many different possible chef-orders are there from the five customers?
• (c,5) In how many of the waiter-orders does each customer order a different kind of roll?
• (d,5) How many possibilities are there for the set of roll types ordered by the five customers? (In our example the set is {A, C, T}.)
• (e,5) How many different waiter-orders could result in the chef-order "two A's, one C, one S, one T"?
• (f,5) How many waiter-orders for the five customers result in all four kinds of roll being ordered at least once? (Hint: Use the result of part (e).)

• Question 6 (30): This problem explores two facts about the number (n choose 3).
• (a,5) For n=3, n=4, and n=5, verify the identity: (n choose 3) = "the sum from 1 to n of (i-1)(n-1)".
• (b,10) Prove the identity in part (a) for all non-negative integers n. (Hint: Consider all the three-element subsets of the set {1,...,n}. Divide them into groups based on their middle element, and count each group.)
• (c,15) Prove by induction that (n choose 3) = "the sum for i from 0 to n-1 of (i choose 2)". (Hint: Remember that you need to prove the base cases for n=0, n=1, n=2, and n=3, then prove that the case for n implies the case for n+1. You may find Pascal's Identity useful -- this says that for any non-negative integers m and k, (m+1 choose k) = (m choose k) + (m choose k-1).)