Solve:
nc6 = nc4
need , i don't want to expand. is there a shortcut method?

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:




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:


Using this:

So, n - 6 = 4
and n = 10.

32 would have an exponent of 3 and 55 would have an exponent of 2.