3.16: Gardening committee, passcodes and combinations identities

A gardening group has 20 members. A committee of 6 members is to be selected. Anwar and Bo belong to the gardening group and at most one of them can be on the committee. How many different committees are possible?

The gate for the garden has a lock with a 6-character passcode made from the letters G, A, R, D, E, N and the digits 0–9. No character may be used more than once. Find the number of possible passcodes that have 4 letters followed by 2 numbers.

Given that $n \geqslant 4$, show that $(n - 3)\dbinom{n}{3} = 4\dbinom{n}{4}$.

Given that $\dbinom{n}{3} = 5n$, where $n \geqslant 3$, show that $n$ satisfies $n^{2} - 3n - 28 = 0$. Hence find the value of $n$.