3. For each of the following pairs of ordered bases ẞ and ẞ' for P2(R), find the change of coordinate matrix that changes ẞ'-coordinates into B-coordinates. (a) = {x2, x, 1} and 3' = {a2x² + a1x+ao, b2x² + b₁x + bo, c2x² + c₁x + co} (b) = {1, x, x2} and ẞ' = {a2x² + a1x+ao, b2x² + b₁x+b0, C2x² + c₁x + co} (c) ẞ= {2x2-x, 3x² + 1, x2} and B' = {1, x, x2} (d) ẞ = {x2x1,x1, x2 +1} and B' = {x²+x+4,4x² 3x + 2,2x²+3} (e) ẞ = 1 {x2x, x²+1, x- 1} and B' = {5x22x-3, -2x²+5x+5, 2x² - x - 3} (f) B = {2x² - x+1,x2+3x-2, -x² + 2x+1} and B' = {9x9, x² + 21x2, 3x²+5x+2}

A bag contains five red marbles, three green ones, one lavender one, three yellows, and two orange marbles. HINT [See Example 7.] How many sets of four marbles include none of the red ones?

(Devore Section 2.3 #30) A friend of mine is giving a dinner party. His current wine supply includes 8 bottles of zinfandel, 10 of merlot, and 12 of cabernet (he only drinks red wine), all from different wineries. (a) If he wants to serve 3 bottles of zinfandel and serving order is important, how many ways are there to do this? (b) If 6 bottles of wine are to be randomly selected from the 30 for serving, how many ways are there to do this? (c) If 6 bottles are randomly selected, how many ways are there to obtain two bottles of each variety? (d) If 6 bottles are randomly selected, what is the probability that this results in two bottles of each variety being chosen? (e) If 6 bottles are randomly selected, what is the probability that all of them are the same variety?