Counting: How many positive integers less than 1000?

How many positive integers less than 1000? a) have distinct digits b) have distinct digits and are even c) divisible by 7 d) divisible by 7 and not 11 c) both 7 and 11 d) either 7 or 11 e) exactly one of 7 or 11 f) neither 7 or 11

Recursive definitions

(See attached file for full problem description) --- Give a recursive definition of a) of the functions max and min so that mx{a1,a2,..an and min {a1,a2,…an} are the maximum and minimum of the n numbers a1,a2,…an respectively b) prove that f12+f22+..fn2 = fnfn+1 whenever n is a positive integer fn is the Fibonacci sequence ...continues

Counting

1. how many license plates can be made using three letters followed by the three digits or four letters followed by two digits 2. how many bit strings of length 10 contain either 5 consecutive 0s or five consecutive 1s.

Show that if 7 integers are selected from the first 10 positive integers there must be at least 2 pairs of these integers with the sum 11. Is the conclusion true if 6 integers are selected instead of 7 How many numbers must be selected from the set {1,3,5,7,9,11,13,15} to guarantee that at least one pair of these numbers add up to 16.

Show that if 7 integers are selected from the first 10 positive integers there must be at least 2 pairs of these integers with the sum 11 Is the conclusion true if 6 integers are selected instead of 7 How many numbers must be selected from the set {1,3,5,7,9,11,13,15} to guarantee that at least one pair of these numbers a ...continues

Counting

A coin is flipped eight times where each flip comes up with either heads or tails how many possible outcomes? a) contain exactly 3 heads b) contain at least 3 heads c) contain the same number of heads n tails.

Counting

7 women and 7 men are in the department of mathematics. How many ways are there to select a committee of 5 members if at least 1 woman and least 1 man must be on the committee?

Proof : Equivalence Relations and Divisibility

Prove 8|5^(n+1) +(2)3^n + 1, n Є N

Equivalence Relations and Classes

For m, n, in N define m~n if m^2 — n^2 is a multiple of 3. (a.) Show that ~ is an equivalence relation on N. (b.) List four elements in the equivalence class [0]. c) List four elements in the equivalence class [1]. (d.) Are there any more equivalence classes. Explain your answer.

Modular Arithmetic Proof

Prove that if m,n Є Z and m ≡ n(mod p) then m^n ≡ n^n(mod p).

Collection of subsets for equivalence relations

Let S be a collection of subsets of X, X = US. [S is not necessarily peicewise disjoint.] xRy if x, y Є S ⊂ S. Is R necessarily an equivalence relation? Show why or why not.