Binomial coefficient
Without replacement, order doesn't matter
From an urn with $n$ different balls, $k$ balls are drawn in succession without replacement.
If the order doesn't matter, the number of possible combinations is $N$:
$N=\frac{n!}{k!\cdot(n-k)!}$ $={n\choose k}$
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$n!$ is called the n-factorial and is the product of all natural numbers from 1 to n.
Example: $5!=1\cdot2\cdot3\cdot4\cdot5$
Example: $5!=1\cdot2\cdot3\cdot4\cdot5$
The term ${n\choose k}$, read: „n choose k“, is called the binomial coefficient.
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Most calculators have their own key for the binomial coefficient: the nCr-key.
Otherwise you always have to enter the fraction with the faculties.
Otherwise you always have to enter the fraction with the faculties.
Example
In the lottery, 6 numbers are drawn from 49 numbers without replacement. The order doesn't matter.
Calculate the different options.
${49\choose 6}$ $=\frac{49!}{6!\cdot43!}$ $=13.983.816$
Only one of these options wins, the probability is:
$\frac{1}{13.983.816}$ $\approx0.000000072$
Example
8 teams take part in a football tournament. How many endgame pairs are possible?
${8\choose 2}=28$