Question

Among nine presidential candidates at a debate, three are Republicans and six are Democrats.

(a) In how many different ways can the nine candidates be lined up?
(b) How many lineups by party are possible if each candidate is labeled either R or D?
(c) For each of the nine candidates, you are to decide whether the candidate did a good job or a poor job; that is, give each of the nine candidates a grade of G or P. How many different "score cards" are possible?

Answer 1

The nine candidates can be lined up in 9! ways. The number of possible lineups by party is '9 choose 3'. The different 'score cards' possible with nine candidates graded either G or P is 2^9.

Step-by-step explanation:

The student's question is a set of combinatorial problems related to presidential candidates and involves principles of permutations and combinations. Here is how the answer to each part breaks down:

Part (a) - Lineup of Candidates

There are nine presidential candidates in total. To line them up, we can use permutations since the order matters. The total number of different ways they can be lined up is 9 factorial (9!).

Part (b) - Lineups by Party

Considering the party labels, we have three Republicans and six Democrats. Here, the arrangement within each party does not matter, only the order of R and D. Therefore, we use combinations, and the total number of lineups by the party would be a combination of 9 by 3, which is also written as 9 choose 3.

Part (c) - Different "Score Cards"

Each candidate can be graded as doing a good job (G) or a poor job (P). With nine candidates, and each one having two possible grades, we have 2 to the power of 9 or 2^9 different scorecards possible.