A husband and wife and their two children line up for a photo.

Please help with the 4 questions below:

9. A husband and wife and their two children line up for a photo. How many ways are there for these four people to line up so that the husband and wife are *not* next to each other? Be sure to show your work.

10.  Suppose you randomly draw two cards from a standard deck without replacement. What is the probability that neither card is the ace of spades?

11. Let V = {a, b, c, d, e} be a vertex set and E = { {a,c}, {b,d}, {c,d}, {d,e}, {e,a}} be the edge set corresponding to V. Explain why the pair (V, E) is or is not a tree.

12. Let vertex sets V1 and V2 be defined by V1= {1, 2, 3} and V2 = {a, b, c}. Let E1 = { { 1, 2}, {2, 3} }, and let E2 = { {a, b},  {b, c} } be the edge sets corresponding to the vertex sets V1 and V2, respectively. Write, as a set of ordered pairs, a function f that is a bijection from V1 to V2, satisfying the following condition: if x and y are elements in V1 such that {x,y} is in E1, then f(x) and f(y) are elements in V2 such that {f(x),f(y)} is in E2, and show that your function f satisfies this condition.

Note: you do not need to show that your function f is a bijection (though it must be, or you won’t get any credit), but you DO need to show that it satisfies the condition “if x and y are elements in V1 such that {x,y} is in E1, then f(x) and f(y) are elements in V2 such that {f(x),f(y)} is in E2.”