Definition of Combinations
A selection of some or all of a number of different objects. It is an un-ordered
collection of unique sizes. The order of occurrence if the objects is not
important.
Formula of Combinations
Example
To simplify this, consider that :
That neat little trick allow us to enormously simplify the combinations
formula :
Types of Combinations
Combinations
with Repetition
Let us
say there are five flavors of ice cream : banana, chocolate, lemon,
strawberry and vanilla.
We can
have three scoops. How many variations will there be?
Let’s use
letters for the flavors : {b, c, l, s, y}.
Example
selections include
- {c, c, c} (3 scoops of chocolate)
- {b, l, v} (one each of banana, lemon and
vanilla)
- {b, v, v} (one of banana, two of vanilla)
Now, I
can’t describe directly to you how to calculate this, but I can show you a special
technique that lets you work it out.
So it is
like we are ordering a robot to get our ice cream, but it doesn’t change
anything, we still get what we want.
We can
write this down as
(arrow
means move, circle means scoop).
In fact
the three examples above can be written like this :
- {c, c, c} (3
scoops of chocolate):
|
|
- {b, l, v} (one
each of banana, lemon and vanilla):
|
|
- {b, v, v} (one
of banana, two of vanilla):
|
|
OK, so
instead of worrying about different flavors, we have a simpler
question :
“How many different ways can we arrange arrows and circles?”
Formula of Combinations in
Repetition
Combinations
without Repetition
This is
how lotteries work. The numbers are drawn one at a time, and if we have
the lucky numbers (no matter what order) we win!
The
easiest way to explain it is to :
- Assume that he order does matter (i.e. Permutations),
- Then alter it so the order does not
matter.
But many
of those are the same to us now, because we don’t care what order!
Example
Let us
say balls 1, 2 and 3 are chosen.
These are
the possibilities :
Order does matter
|
Order doesn't matter
|
1 2 3
1 3 2
2 1 3
2 3 1
3 1 2
3 2 1
|
1 2 3
|
So, the
permutations will have 6 times as many possibilities.
In fact
there is an easy way to work out how many ways “1 2 3” could be placed in
order, and we have already talked about it.
The
answer is :
So we
adjust our permutations formula to reduce it by how many ways the
objects could be in order (because we aren’t interested in their order any
more) :
Formula of Combinations
without Repetition
That
formula is so important it is often just written in big parentheses like this :
And is
also known as the Binomial Coefficient.
Notation
As well
as the “big parentheses”, people also use these notations :
Example
So, our
pool ball example (now without order) is :
Or we
could do it this way :
It is
interesting to also note how this formula is nice and symmetrical :
In other
words choosing 3 balls out of 16, or choosing 13 balls out of 16 have the same
number of combinations.
Pascal’s
Triangle
Here is
an extract showing row 16 :
Here are the examples with solutions
:
Example 1
Rohan has 3 shirts and 2 pants, in how many are the
combinations possible?
He can select any shirt from 3 shirts and any pant from 3
pants.
Example 2
Notice
how the cancellation occurs, leaving only 2 of the factorial terms in the
numerator. A pattern is emerging when finding a combination such as the one
seen in this problem, the second value (2) will tell you how many of the
factorial terms to use in the numerator, and the denominator will simply be the
factorial of the second value (2).
Example 3
There are
12 boys and 14 girls in Mrs. Schultzkie’s math class. Find the number of ways
Mrs. Schultzkie can select a team of 3 students from the class to work on a
group project. The team is to consist of 1 girl and 2 boys.
Order, or
position is not important. Using the multiplication counting principle,
References :
