Binomial Coefficient/Examples

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Examples of Binomial Coefficients

$2$ from $5$

The number of ways of choosing $2$ objects from a set of $5$ is:

$\dbinom 5 2 = \dfrac {5 \times 4} {2 \times 1} = \dfrac {5!} {2! \, 3!} = 10$


$3$ from $7$

The number of ways of choosing $3$ objects from a set of $7$ is:

$\dbinom 7 3 = \dfrac {7 \times 6 \times 5} {3 \times 2 \times 1} = \dfrac {7!} {3! \, 4!} = 35$


$3$ from $8$

The number of ways of choosing $3$ objects from a set of $8$ is:

$\dbinom 8 3 = \dfrac {8 \times 7 \times 6} {3 \times 2 \times 1} = \dfrac {8!} {3! \, 5!} = 56$


$4$ from $52$

The number of ways of choosing $4$ objects from a set of $52$ (for example, cards from a deck) is:

$\dbinom {52} 4 = \dfrac {52 \times 51 \times 50 \times 49} {4 \times 3 \times 2 \times 1} = \dfrac {52!} {48! \, 4!} = 270 \, 725$


Number of Bridge Hands

The total number $N$ of possible different hands for a game of bridge is:

$N = \dfrac {52!} {13! \, 39!} = 635 \ 013 \ 559 \ 600$