3.6. Basic Counting Rules http://www.ck12.org
n= 13
r= 3Using the permutation formula,
Prn=n!
(n−r)!=
13!
( 13 − 3 )!
= 1716
There are 1716 different slates of officers.
Notice that in our previous examples, the order of people or objects was taken into account. What if the order is not
important? For example, in the previous example for electing three officers, what if we wish to choose 3 members
of the 13−member board to attend a convention. Here, we are more interested in the group of three but we are not
interested in their order. In other words, we are only concerned with different combinations of 13 people taken 3
at a time. The permutation rule will not work here since order is not important. We have a new formula that will
compute different combinations.
Counting Rule for Combinations
The number of combinations ofnobjects takenrat a time is
Crn=n!
r!(n−r)!It is important to notice the difference between permutations and combinations. When we considergrouping and
order, we use permutations. But when we considergrouping with no particular order, we use combinations.
Example:
Back to our example above. How many different groups of three are there, taken out of 13 people?
Solution:
As explained in the previous paragraph, we are interested in combinations rather than permutations of 13 people
taken 3 at a time. We use the combination formula
Cnr=
n!
r!(n−r)!
C^133 =13!
3!( 13 − 3 )!
=
13!
3! 10!
= 286
There are 286 different groups of 3 to go to the convention.
In the above computation you can see that the difference between the formulas fornCrandnPris in the factorr!
in the denominator of the fraction. Sincer! is the number of different orders ofrthings, and combinations ignore
order, then we divide by the number of different orders.
Example: