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The Binomial Coefficients ⁿC_r

Symmetry ⁿC_r = ⁿC_{n - r}
Symmetry is a way of providing a shortcut when r is a large number. An example could be choosing 11 people (r) from 15 (n) - you can either name the 11 you've selected or the 4 you've rejected. So every choice you have of r objects included in a selection from n distinct objects corresponds to a choice of (n - r) objects which are excluded. So basically, ⁿC_r = ⁿC_{n - r}. You can also see the list of values of ⁿC_r for any particular value of n is unchanged by being reversed. E.g. when n = 6, the list is still 1, 6, 15, 20, 15, 6, 1.

Addition ^{n + 1}C_{r + 1} = ⁿC_r + ⁿC_{r + 1}
For this we need to have a look at Pascal's triangle:

If we look at the 15 on the 6th row, we can see that the two numbers above it add together to give the value, 10 + 5. We know that 10 and 5 are both ⁿC_r themselves, and can be called ⁵C₁ and ⁵C₂. So, we can say that ⁵C₁ + ⁵C₂ = ⁶C₂ (the ⁿC_r value of 15. We could use this method to build up a table of ⁿC_r values without calculating much. Lets give an example that X is either included or not. If it's included, you have to choose r more objects from the other n objects. This can be done in ⁿC_r ways. If it's not included, you have to choose all r + 1 objects from another n objects - ⁿC_{r + 1} ways. These two cases cover all possibilities so ^{n + 1}C_{r + 1} = ⁿC_r + ⁿC_{r + 1}.