The New Math SAT Game Plan is a fantastic book.

Amazing.

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.

Fantastic!

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.

Seriously.

*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.