First, a few practice questions. Remember — no calculator!

**Q1**. A radio station has to choose three days of the seven in a week to broadcast a certain program, and that set will repeat each week. The program can be broadcast equally on any of the seven weekdays — weekdays vs. weekends don't matter at all — nor does it matter whether the days the program airs are adjacent or not. Absolutely any three of the seven weekdays can be chosen. How many different three-day combinations of the seven weekdays can be constructed?

(A)

(B)

(C)

(D)

(E)

**Q2**. Claudia can choose any two of four different candles and any

(A)

(B)

(C)

(D)

(E)

**Q3**. A newly-wed couple is using a website to design an eBook Wedding Album to distribute to their friends and families. The template they have chosen has places for

(A)

(B)

(C)

(D)

(E)

**Q4**. From a total of

(A)

(B)

(C)

(D)

(E)

In this post, we'll discuss how to handle questions like this — without a calculator.

Mathematically, a

In a previous post about combinations, I give the following formula for

where the exclamation point ("**!**") is the factorial symbol —

So, it turns out, there are twenty ways to pick a set of three items from a pool of six unique items. That's one way to calculate nCr, but it's not the only way.

The mathematician and philosopher Blaise Pascal (1623 – 1662) created a magical triangular array of numbers known now as Pascal's Triangle:

How does this pattern work? Well, of course, the edges are diagonals of

Despite its relatively easy origins, Pascal's Triangle is a treasure trove of miraculous mathematical properties. Most relevant for us right now is: Pascal's Triangle is, among other things, an array of all possible

In that definition, we have to be careful — we have to start counting at zero instead of one. The top

That makes sense: if we have

When you have to figure out

Things get even more interesting when we move to the next diagonal in, shown in green here:

These numbers, the set of the second entries in each row, are the triangular numbers. Among other things, the second entry in the

The formula for this is:

Because we have a formula, we can calculate this for much higher numbers. For

Notice that the symmetry of Pascal's Triangle also provides tremendous insight into the nature of the nCr numbers. First of all, in any row, the second entry, the triangular number in that row, must be equal to the third-to-last entry of the row, that is, the

Thus, via the triangular numbers, we have a formula, not only for the second entry of each row, but also for the third-to-last entry of every row. Thus, it's very easy to figure out the first three or last three numbers in any row. More generally, symmetry guarantees that:

If you think about combinations this makes sense: if we have a pool of

**Q1**. Behind the story, we are really being asked to evaluate

**Answer = D**

**Q2**. For this one, we have to use the Fundamental Counting Principle (FCP) as well as information about combinations. For the flowers, we want

That's the number of flower combinations. For the candles,

Now, by the FCP, we multiply these for the total number of centerpiece arrangements:

**Answer = A**

**Q3**. For the large photos, we need

For the smaller photos, we need

Now, by the FCP, we just multiply these: total number of possible

**Answer = B**

**Q4**. From a total of

In a way, this is just a question of "and" or "or." Make sure that we don't confuse it with probability, though, which has very different rules for "and" and "or."

In this question, we're looking for the number of different ways we can choose both

Here's a couple of simple examples to illustrate what I mean by this:

If I have an a peach, a plum, an apricot, an orange, a lime, and a lemon, how many ways can I pick two pieces of fruit with pits OR two pieces of citrus fruit? Well, there are

- peach, plum
- peach, apricot
- plum, apricot

- orange, lemon
- orange, lime
- lemon, lime

From that same group, how many ways can I pick two pieces of fruit with pits AND two pieces of citrus fruit? This gives more possibilities.

- peach, plum, orange, lemon
- peach, plum, orange, lime
- peach, plum, lemon, lime

- peach, apricot, orange, lemon
- peach, apricot, orange, lime
- peach, apricot, lemon, lime

- plum, apricot, orange, lemon
- plum, apricot, orange, lime
- plum, apricot, lemon, lime

That makes

Really, we found that there are

This is the same as the boys and girls situation. If we asked how many ways you can make a group of **and** girls, and so we multiply.

**Answer = C**