The use of combination in statistics Why can't we use 0 in combination?
Like for example, count the probability of a person wins at least one prize if he buys 10 out of 1000 tickets in which 5 tickets will be drawn to be the winner. The solution shows that the P is 1- 5C990/5C1000.
Why cant I use 1 - 0C10/5C1000 instead? Thank you
Actually, the correct calculation is $$P=1-\frac{\binom{10}{0}\binom{990}{5}}{\binom{1000}{5}}$$
Representing that out of the $5$ cards picked from $1000$, $0$ winners are picked from the $10$ tickets that he bought, and  $5$ winners are picked from the $990$ he didn't
But $\binom{10}{0}=1$, so it is just left out of the solution.
For a calculation of the form $$\frac{\prod_{j\in\Bbb N}\binom{x_j}{y_j}}{\binom xy}$$ to be accurate in representing a probability we require that $\sum_{j\in \Bbb N} y_j= y$
Actually, the correct calculation is $$P=1-\frac{\binom{10}{0}\binom{990}{5}}{\binom{1000}{5}}$$
Representing that out of the $5$ cards picked from $1000$, $0$ winners are picked from the $10$ tickets that he bought, and  $5$ winners are picked from the $990$ he didn't
But $\binom{10}{0}=1$, so it is just left out of the solution.
For a calculation of the form $$\frac{\prod_{j\in\Bbb N}\binom{x_j}{y_j}}{\binom xy}$$ to be accurate in representing a probability we require that $\sum_{j\in \Bbb N} y_j= y$