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.

When rates are expressed as a quantity of 1, such as 2 feet per second or 5 miles per hour, they are called unit rates. if you have a multiple-unit rate such as 120 students for every 3 buses, and want to find the single-unit rate, write a ratio equal to the multiple-unit rate with 1 as the second term.