Symmetry Rule for Binomial Coefficients/Examples/8 choose 6
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Example of Use of Symmetry Rule for Binomial Coefficients
Let $N$ be the number of ways a team of $6$ people may be selected from a pool of $8$.
Then:
- $N = 28$
Proof
We have:
| \(\ds N\) | \(=\) | \(\ds \dbinom 8 6\) | Cardinality of Set of Subsets | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dbinom 8 2\) | Symmetry Rule for Binomial Coefficients | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {8 \times 7} {2 \times 1}\) | Definition of Binomial Coefficient | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac {56} 2\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 28\) |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Exercises $\text I$: $4$