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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

nCr       or       nCr       or       C(n, r)       or       

and reads 'from n Choose r'.

 


Explanation

It equals

what you can also write as

 


Example 1

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 per 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 applies

,   ,   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.

 


History

These numbers were also called Pascal numbers, in tribute to the French mathematician Blaise Pascal (1623 - 1662).


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