Henry Ernest Dudeney/Modern Puzzles/167 - A General Election/Solution

by : $167$

 * A General Election
 * In how many different ways may a Parliament of $615$ members be elected if there are only $4$ parties:
 * Conservatives, Liberals, Socialists, and Independents?
 * You see you might have $\text C. 310$, $\text L. 152$, $\text S. 150$, $\text I. 3$;
 * or $\text C. 0$, $\text L. 0$, $\text S. 0$, $\text I. 615$;
 * or $\text C. 205$, $\text L. 205$, $\text S. 205$, $\text I. 0$; and so on.
 * The candidates are indistinguishable, as we are only concerned with the party numbers.

Solution

 * $39 \, 147 \, 416$ different ways.

Proof
According to :


 * The general solution is as follows.
 * ''Let $p =$ parties and $m =$ members.
 * Then $C^{p - 1}_{m + p - 1} =$ number of ways.

In the above, $C^{p - 1}_{m + p - 1}$ is the binomial coefficient $\dbinom {p - 1} {m + p - 1}$.