Symmetry Rule for Binomial Coefficients/Examples
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Examples of Use of Symmetry Rule for Binomial Coefficients
$11$ choose $8$
Consider the binomial coefficient $\dbinom {11} 8$.
This can be calculated as:
- $\dbinom {11} 8 = \dfrac {11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4} {8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$
which is unwieldy.
Or we can use the Symmetry Rule for Binomial Coefficients, and say:
- $\dbinom {11} 8 = \dbinom {11} {11 - 8} = \dbinom {11} 3$
and calculate it as:
- $\dbinom {11} 3 = \dfrac {11 \times 10 \times 9} {3 \times 2 \times 1} = \dfrac {990} 6 = 165$
which is far less trouble.
$8$ choose $6$
Let $N$ be the number of ways a team of $6$ people may be selected from a pool of $8$.
Then:
- $N = 28$