In a production of west side story, eight actors are considered for the male roles of Tony, Riff, and Bernardo.
a.) is this permutation or combination?
b.) in how many ways can the director cast the male roles?
If the roles can be changed among the three selected actors, then it is a combination. If they cannot, then it is a permutation.
Therefore a permutation is usually a bigger number than a combination, because to get the combination, we divide the permutation by the number of arrangements of the selected candidates.
Permutation: P(n,r)=n!/(n-r)!
Combination: C(n,r)=n!/((n-r!)r!)
b.) 8P3 = 336
a.) This is a combination because the order of the male roles does not matter. It's not like they're dealing with a Tony sandwich where the placement of each layer matters.
b.) To calculate the number of ways the director can cast the male roles, we can use a combination formula. Since there are 8 actors considered for 3 roles, it would be 8 choose 3. This can be calculated as:
8! / (3!(8-3)!) = 8! / (3!5!) = (8 × 7 × 6) / (3 × 2 × 1) = 56
So, there are 56 ways in which the director can cast the male roles. That's a lot of Tony, Riff, and Bernardo combinations to choose from!
a.) This is a permutation.
b.) Since there are 8 actors being considered for the 3 male roles, the director can cast the male roles in 8P3 ways.
The formula for permutations is nPr = n! / (n - r)!, where n is the total number of options and r is the number of selections.
Therefore, the number of ways the director can cast the male roles in this scenario is 8! / (8 - 3)! = 8! / 5! = 8 * 7 * 6 = 336 ways.
a.) This is a combination problem because the order in which the actors are cast for the roles does not matter.
b.) To find the number of ways the director can cast the male roles, we need to use the combination formula. Since there are 8 actors considered for the 3 male roles, we need to calculate the number of combinations of 8 actors taken 3 at a time.
The combination formula is given by:
C(n, r) = n! / (r!(n - r)!)
Where n is the total number of actors (8) and r is the number of roles to be cast (3).
So, in this case, the number of ways the director can cast the male roles is:
C(8, 3) = 8! / (3!(8 - 3)!)
= 8! / (3! * 5!)
= (8 * 7 * 6) / (3 * 2 * 1)
= 56
Therefore, the director can cast the male roles in 56 different ways.