Binomial Coefficient/Examples/2 from 5
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Example of Binomial Coefficient
The number of ways of choosing $2$ objects from a set of $5$ is:
- $\dbinom 5 2 = 10$
Proof
| \(\ds \dbinom 5 2\) | \(=\) | \(\ds \dfrac {5!} {2! \, 3!}\) | Definition of Binomial Coefficient | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {5 \times 4 \times 3 \times 2 \times 1} {\paren {2 \times 1} \paren {3 \times 2 \times 1} }\) | Definition of Factorial | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {5 \times 4} {2 \times 1}\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {20} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 10\) |
$\blacksquare$
Sources
- 1964: A.M. Yaglom and I.M. Yaglom: Challenging Mathematical Problems With Elementary Solutions: Volume $\text { I }$ ... (previous) ... (next): Problems