Symmetry Rule for Binomial Coefficients/Examples/11 choose 8

Example of Use of Symmetry Rule for Binomial Coefficients

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.

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text I$. Algebra: Permutations and Combinations: Two important relations