to summarize;
Combination: a combination is an unordered collection of unique elements
ie: numbers, variables
ie: numbers, variables
- the order of the combination is not important
- two lists with the same elements in a different order are considered to be the same combination
- the elements cannot be repeated, they must appear once, thus making them "unique"
- combinations are defined by the elements contained in them
http://www.dictionary.com/ states that a combination is:
the arrangement of elements into various groups without regard to their order in the group.
the arrangement of elements into various groups without regard to their order in the group.
This is a link I found on http://www.reference.com/ to
Many Common types of permutation and combination math problems, with detailed solutions
I'm not sure how useful it will be, I browsed through and it seemed to have potential, so hopefully it provides some useful information.
Many Common types of permutation and combination math problems, with detailed solutions
I'm not sure how useful it will be, I browsed through and it seemed to have potential, so hopefully it provides some useful information.
This link leads to some random school districts exam prep page, it looks very very useful, it has lessons and practice and formulas, yay!
http://regentsprep.org/regents/math/math-topic.cfm?TopicCode=combin
http://regentsprep.org/regents/math/math-topic.cfm?TopicCode=combin
2 comments:
A combination is an unordered collection of unique elements (an ordered collection is called a permutation).
This is what I found on www.themathpage.com about combinations:
In permutations, the order is all important -- we count abc as different from bca. But in combinations we are concerned only that a, b, and c have been selected. abc and bca are the same combination.
Here are all the combinations of abcd taken three at a time:
abc abd acd bcd.
There are four such combinations. We call this
The number of combinations of 4 things taken 3 at a time.
We will denote this number as 4C3. In general,
nCk = The number of combinations of n things taken k at a time.
Now, how are the number of combinations nCk related to the number of permutations, nPk ? To be specific, how are the combinations 4C3 related to the permutations 4P3?
Since the order does not matter in combinations, there are clearly fewer combinations than permutations. The combinations are contained among the permutations -- they are a "subset" of the permutations. Each of those four combinations, in fact, will give rise to 3! permutations:
abc abd acd bcd
acb adb adc bdc
bac bad cad cbd
bca bda cda cdb
cab dab dac dbc
cba dba dca dcb
Each column is the 3! permutations of that combination. But they are all one combination -- because the order does not matter. Hence there are 3! times as many permutations as combinations. 4C3 , therefore, will be 4P3 divided by 3! -- the number of permutations that each combination generates.
4C3 =
3! = 4· 3· 2
1· 2· 3
Notice: The numerator and denominator have the same number of factors, 3, which is indicated by the lower index. The numerator has 3 factors starting with the upper index and going down, while the denominator is 3!.
In general, nCk =
k! .
nCk = n(n − 1)(n − 2)· · · to k factors
k!
So to conclude, in combinations we are concerned only that a, b, and c have been selected. abc and bca then are the same combination. Also when working with combinations of 4 things taken 3 at a time, we will denote this number as 4C3. In general, nCk = The number of combinations of n things taken k at a time.
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