
Permutations and Combinations
As most high school students correctly suspect, SAT testmakers
confine themselves in a dungeonlike environment with a few unfortunate
teenage students, poking and prodding them with ideas for the upcoming exam.
In this inquisition atmosphere, the darkest region, from which few—if
any—students escape, is the Permutations and Combinations Iron Maiden.
It is the most gruesome, horrific, and sadistic topic that appears
on the SAT Math section. For this reason, permutations and combination
items are always considered difficult and, therefore, always appear
toward the end of a Math set.
Factorials
The factorial of a number, represented by n!,
is the product of all the natural numbers up to and including n.
So if you were asked to find the factorial of 4, it would look like
this:
The factorial of a number is useful because it expresses
the number of ways that n elements of a group can
be ordered. So if you had four snow globes from four different cities
and you wanted to know how many different ways they could be arranged
on your window sill, the answer would be 4!, or .
Permutations
Permutations also deal with the seemingly
countless ways that a certain number of things can be ordered. Let’s
say we have five students competing in the Olympic sport of underwater
basket weaving (this sport doesn’t get much airplay). The judges
feel that every competitor is a winner in his or her own way and
thus ordered five different medals for the five competitors: gold
for first, silver for second, bronze for third, iron for fourth,
and copper for fifth. How many different arrangements of medal winners
could there be? This question is exactly like the snow globe question.
We simply take the factorial of 5:
So there are 120 different arrangements of placing the
competitors from first to fifth place.
But let’s say that the Olympic Games Committee finds out
what the judges are up to and immediately puts a stop to this unfair
use of medals, quickly reinstating the standard gold, silver, and
bronze medals for the top three competitors, while the last two
competitors receive nothing. How many different arrangements of
the five competitors could receive the three medals? Now we are
dealing with a permutation, which means we need to trot out the
following formula:
Now, n is the total number of elements
(people, snow globes, etc.) that we are dealing with, and r is
the size of the subgroup that we are fitting our total elements
into—in this case, the subgroup is the number of medals. To find
the different arrangements of the five competitors receiving the
three medals, we plug 5 and 3 into the equation:
Notice that in permutations, order matters. In
other words, if first and third positions changed with
each other and the rest of the order stayed the same, it would still
be considered a different arrangement. This is important in order
to distinguish between permutations and combinations.
Combinations
A combination is an unordered grouping of
a set. An excellent example of a combination scenario is when you
and your friends order pizza. Whether you order pepperoni,
mushrooms, and onions, or you order mushrooms, onions, and pepperoni
doesn’t matter—it’s still the same pizza. The most important thing
to remember about combinations is order does not matter.
Because the order of the subgroup doesn’t matter, the
combination solutions will be fewer than the permutation solutions
and will be expressed by the following formula:
The variable n is the total
number of elements, and r is the number placed
in the subgroups. Let’s elaborate on our previous example with the
Olympic basket weavers and ask how many different combinations of
gold, silver, and bronze competitors are possible. Because it no
longer matters who wins the particular medals, we will use the combination
formula:
There are 10 different combinations of medal
winners, or 10 different groups of 3 that would win either gold,
silver, or bronze medals, as opposed to 60 different arrangements
of individual gold, silver, and bronze winners.
That’s it for DS&P. Now that you’ve mastered these
concepts, it’s time to show you the best way to put this knowledge
to use.
