Symmetry Rule for Binomial Coefficients/Examples

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.

$7$ from $10$

The number of ways of choosing:

$7$ objects from a set of $10$

is the same as the number of ways of choosing:

$3$ objects from a set of $10$.

Hence:

$\dbinom {10} 7 = \dbinom {10} 3 = \dfrac {10 \times 9 \times 8} {3 \times 2 \times 1} = \dfrac {720} 6 = 120$

$8$ choose $6$

Let $N$ be the number of ways a team of $6$ people may be selected from a pool of $8$.

Then:

$N = 28$