Problem
I'm told that a bag of marbles contains $10$ red and $5$ white marbles. Each of the red and white marbles is numbered such that they are distinct within their own groups, i.e.
$R = \{a, b, c, d, e, f, g, h, i, j\}$
$W = \{1, 2, 3, 4, 5\}$
I'm asked to calculate how many combinations there are if I am to select $4$ marbles from the bag but at least one of them must be a red marble.
Attempt
If there is at least $1$ marble, then that is $10$ choose $1$, which is just $10$. Afterwards, we are left with $14$ marbles in total and need to choose $3$ of them, which is $364$. Multiplying by $10$, I get $3,640$, but the answer key says it's supposed to be $1,360$.
Where did I go wrong?
Edit: I tried using the counting by complement rule and got the right answer, but I'm still curious what's wrong with this approach.