Final answer:
To assign twelve different roles to twelve actors, we calculate the permutations using 12 factorial, resulting in 479,001,600 ways. When there are only seven roles for twelve actors, we use the permutation formula 12P7, equating to 5,040 ways. Thus, the correct answer to both parts of the question is option (c).
Step-by-step explanation:
When dealing with the assignment of roles to actors, we tackle two different problems: permutation and combination, depending on whether the number of roles is equal to or less than the number of actors. For part (a) where there are twelve different roles for twelve actors, we use permutation since the order in which we assign the roles to different actors matters. This can be calculated using a factorial, which for twelve items is 12 factorial (12!).
The formula for the permutation of n items taken n at a time is n!, so for part (a) with twelve roles, it's 12! which equals 479,001,600 ways to assign the roles.
For part (b), where there are only seven roles but twelve actors to choose from, we use the permutation formula nPk which represents the number of permutations of n items taken k at a time. This is calculated as n!/(n-k)! For part (b), it's 12P7, giving us 12!/(12-7)! which simplifies to 12!/5! and equals 5,040 different ways.
Hence, the correct answer is choice (c): (a) 479,001,600, (b) 5,040.