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Combinations
The number of combinations to choose r elements from a set of n elements, without paying attention to the sequence, can be written as
and reads 'from n Choose r'.
Calculation formula
It equals
what you can also write as
Explanation
We take the letters ABCD and start playing with them. Let us see how many different combinations of 2 letters you can create with these 4 letters. If the sequence of the letters plays no role, we have 6 combinations
AB = BA
AC = CA
AD = DA
BC = CB
BD = DB
CD = DC
We call this 'from 4 choose 2' and calculate
We continue, and look for the combinations of 3 letters out of the 4 letters. As the sequence of the letters plays no role, we have 4 combinations
ABC = ACB = BAC = BCA = CAB = CBA
ABD = ADB = BAD = BDA = DAB = DBA
ACD = ADC = CAD = CDA = DAC = DCA
BCD = BDC = CBD = CDB = DBC = DCB
We call this 'from 4 choose 3' and calculate
If we take all 4 letters there remains ony 1 combination
ABCD = DCBA … and so on
We call this 'from 4 choose 4' and calculate that also
It is determined by definition that 0! ≝ 1, and that helps us here. For completeness we also take 1 letter, and get 4 combinations
A
B
C
D
We call this 'from 4 choose 1' and calculate that as
If we take no letter all, we call this 'from 4 choose 0' and calculate that
Why we do this is not clear now, but the formula can handle it. There is one thing we have not looked at yet. Consider the calculations we encountered:
They stood for
That is amazing. First the numbers increase, and then they decrease again. And that is logical if you look at the two factors in the denominator of the formula. If the first factor increases, the second factor decreases accordingly. You can easily see that in general holds
,
, and
We check now a special case, where n tends toward infinity
In the fraction, the numerator and denominator it deals about "the same Infinity", and that disappears. So it is all quite robust.
Example 1
You can see that 5 nCr 2 = 10, because
HistoryThese numbers were also called Pascal numbers, in tribute to the French mathematician Blaise Pascal (1623 - 1662). |