Last night I read the section on counting.
Counting has been a massive struggle for me. I worked my way through Dolciani's chapter on combinations and permutations, did all the exercises, and ended up pretty much where I started out: basically, being "counting" blind: all the problems look alike. I can't tell the difference amongst them, and I can't tell when and where I would do what or why. I have been utterly flummoxed. *
Last night, reading Keller, everything clicked.** From one moment to the next, I abruptly understood why all the problems look alike (to me) and what the solutions have in common logically. I finished up Keller's 5-page explanation and did all of his exercises quickly and correctly. Easy-peasy.
Later on, I'll try Dolciani's exercises and see how I fare.
Here's how Keller explains the counting principle:
Your favorite restaurant offers a combo-meal. You get to pick one each from a menu of 6 sandwiches, 4 side salads, 5 beverages, 10 desserts and 3 sounvenir toys. You decide to eat at this restaurant once every day, ordering a sandwich, salad, beverage, dessert and toy every time, until you have had every possible combination. To the nearest whole number, how many years will it take you??
Do NOT attempt to list all of the combinations. Instead, learn the Counting Principle:
In any situation where you are faced with a series of decisions keep asking yourself:
"Now, how many choices do I have?"
until the last decision has been made. Then to find the overall number of combinations, you multiply together all the numbers of choices you had for each decision.
So in the example I have given you, you have to choose your sandwich from 6 choices, then your salad from 4 choices, your beverage from 5, your dessert from 10 and your toy from 3. And then you are done making decisions. So you multiply and find that there are
6 x 4 x 5 x 10 x 3 =3600 combinations. And then we can divide by 365 days in a year and find that it would take just under 10 years to order every combination. That's a long time but don't be surprised. When you have lots of decisions, or lots of options you get big numbers.
The example I just gave you is one of the easiest kinds of counting problems you'll see on an SAT. Many of the other varieties are a little harder to recognize and a little trickier to answer, as the next few examples will show.
After this, Keller shows how the solution to the problem where you've got a family sitting in a row and the mom and dad have to sit on the end chairs while the four kids can sit on any of the in-between chairs (and how many combinations is that???!!!) follows exactly from the solution to the how-many-combinations-in-the-restaurant problem.
I am SO happy to know how many ways a family of 6 can sit in 6 chairs with both parents occupying the end chairs.
Now how many choices do I have?
* I have yet to use either of the two resources you all left for me: the Arlington Agebra Project and a web page created by a math professor who may have sent me the link via email (I don't remember - !). Since I don't remember, I won't link here.
** Well...not everything. Still having trouble with the SAT counting problem that nearly did me in last summer - but I now understand the first part of the problem.