Be Thankful For Math! - The Solutions
November 25, 2002
Since Grady can’t read, we can assume that each person’s name card has an equal probability of being placed at any of the table settings. Therefore, the probability that Grandma Sherry is placed at one of the two seats on the shorter sides of the table is 2/8, which is 1/4, as a common fraction.
Since everyone is taking turkey, we really just have to count how many ways a person can choose from P (potatoes), C (carrots), G (green beans), S (cranberry salad) and D (dinner rolls). A person could choose none of the side dishes – this is one combination. A person could just choose one of the side dishes. There are five to choose from, so that’s five ways to choose just one side dish. Now, choosing two side dishes gets a little more complicated. Here are all of the possibilities for two side dishes: PC, PG, PS, PD, CG, CS, CD, GS, GD and SD. That makes ten ways to choose two side dishes. Notice this can also be done with the formula for “5 choose 2” which is (5!) ÷ [(2!) × (5-2)!] = 5! ÷ [(2!)(3!)] = 10. Now, to choose three of the five side dishes seems like it will be even more complicated than choosing two, but using the formula makes this pretty easy. Also notice, that for each pair of side dishes that we chose before, we left a grouping of three side dishes out. If there are ten distinct pairs of side dishes, then there must also be ten distinct trios of side dishes! Now, how many ways can you choose four side dishes? Think of it this way, if you want 4 side dishes, then you’re leaving one out. How many ways can you leave just one out? Five ways, since there are five options! And of course, like Grandpa Curt, a person could choose all five side dishes, for one last combination. That’s a total of 1 + 5 + 10 + 10 + 5 + 1 = 32 combinations of side dishes a person could choose to go with the turkey!
Since there is 1/3 of the pie left, that means 2/3 of the pie was cut into the eight equal pieces. A sector that is two-thirds of a round pie has a central angle measure of (2/3)(360) = 240 degrees. Split evenly into eight pieces of pie, this would make the central angle of each person’s slice of pie 30 degrees.