Yes, there is.
Of course, the longer way is to formulate the combinations formula, but we need to understand what nCr actually represent in order to understand the shortcut.
For a binomial expansion (1 + x)ⁿ, we always have nC0, nC1, etc. at the start of each term because they represent coefficients of a binomial expansion. This is the core foundation of a binomial expansion. However, using this property, we also know that for a binomial expansion, there lies another property:
the Binomial coefficient symmetry:
This is a property in every binomial expansion, and we can see this if we were to expand the (1 + x)ⁿ from n = 1 to n = 3. This rule shows that, from the middle term, each term before and after it is equal. Let's use this rule to demonstrate the property:
[tex](1 + x)^{2} = \boxed{1} \cdot 1 + 2x + \boxed{1} \cdot x^{2}[/tex]
[tex](1 + x)^{3} = 1 \cdot 1 + 3x + 3x^{2} + 1 \cdot x^{3}[/tex]
In this case, we have an even number of terms, and thus, the middle coefficient is hidden as 0. This still applies.
So, we can conclude:
[tex]^{n}C_r = ^{n}C_{n - r}[/tex]
Using this:
[tex]^{n}C_6 = ^{n}C_{n - 6} = ^{n}C_4[/tex]
So, n - 6 = 4
and n = 10.
