# Practice Exam for First Midterm

### Directions:

• There are seven problems on pages 2-7, for 100 total points. Probable scale is A=90, C=60.
• The first five questions are true/false, with five points for the correct boolean answer and up to five for a correct justification.
• Question 6 has 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.

```  Q1: 10 points
Q2: 10 points
Q3: 10 points
Q4: 10 points
Q5: 10 points
Q6: 30 points
Q7: 20 points
Total: 100 points
```

• Question 1 (10): True or false with justification: The number of binary strings of length n, with exactly k ones, is (n choose k).

• Question 2 (10): True or false with justification: There are more than 50 different four-element subsets of the set {a,b,c,d,e,f,g,h}.

• Question 3 (10): True or false with justification: If f is a function defined by the rules f(0) = 393 and f(n+1) = f(n) + 146 for all natural numbers n, then f(n) is odd for all natural numbers n.

• Question 4 (10): True or false with justification: If the function T satisfies the rules T(0) = 0, T(1) = 0, and T(n+1) = T(n) + O(n) for all positive integers n, then T(n) = O(n log n).

• Question 5 (10): True or false with justification: If A, B, and C are any three finite sets, then the size of the direct product A × (B ∪ C) is at least |A||B| + |A||C|.

• Question 6 (30): A package of Chewy Candies contains ten candies, each of which is either Cherry, Lemon, Orange, or Strawberry.

• (a,5) How many different possible packages are there if we do consider the order in which the candies occur? (So a package might be described by the string "CCOLCLSCCO".)

• (b,5) How many possibilities are there for the set of flavors that occur in the package?

• (c.10) How many of the answers to (a) contain at least one Cherry and at least one Lemon?

• (d,10) How many different possible packages are there if we do not consider the order in which the candies occur? (So a package might be described as "five cherry, two lemon, two orange, one strawberry".)

• Question 7 (20): Remember that Pascal's Identity says that for any natural numbers a and b, (a+1 choose b) = (a choose b) + (a choose b-1). Let k be any natural number. Prove that for all natural numbers n, (n+k+1 choose k+1) is equal to the sum, for i from 0 to n, of (k+i choose k). (Hint: Use ordinary induction on n, starting with n=0.)