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:
- $\dbinom 5 2 = 10$
$5$ from $2$
- $\dbinom 2 5 = 0$
$2$ from $-5$
- $\dbinom {-5} 2 = 15$
$5$ from $-2$
- $\dbinom {-2} 5 = -6$
$3$ 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:
- $N = \dfrac {52!} {13! \, 39!} = 635 \ 013 \ 559 \ 600$