Answer: 1) 6300 ways
2) 2520 ways
3) 0.067
Step-by-step explanation:
For this problem, assume 9 males audition, one of them being Winston, 5 females audition, one of them being Julia, and 6 children audition. The casting director has 3 male roles available, 1 female role available, and 2 child roles available.
How many different ways can these roles be filled from these auditioners?
Available: 9M and 5F and 6C
Cast: 3M and 1F and 2C
As it is not ordered: C₉,₃ * C₅,₁ * C₆,₂
C₉,₃ = 9!/3!.6! = 84
C₅,₁ = 5!/1!.4! = 5
C₆,₂ = 6!/2!.4! = 15
C₉,₃ * C₅,₁ * C₆,₂ = 84.5.15 = 6300
How many different ways can these roles be filled if exactly one of Winston and Julia gets a part?
2 options Winston gets or Julia gets it:
1) Winston gets it but Julia no:
8 male for 2 spots
4 females for 1 spot
6 children for 2 spots
C₈,₂ * C₄,₁ * C₆,₂
C₈,₂ = = 8!/2!.6! = 28
C₄,₁ = 4!/1!.3! = 4
C₆,₂ = 6!/2!.4! = 15
C₈,₂ * C₄,₁ * C₆,₂ = 28.4.15 = 1680
2) Julia gets it but Winston does not
8 male for 3 spots
1 female for 1 spot
6 children for 2 spots
C₈,₃ * C₆,₂
C₈,₃ = 8!/3!.5! = 56
C₆,₂ = 6!/2!.4! = 15
C₉,₃ * C₆,₂ = 56.15 = 840
1) or 2) = 1) + 2) = 1680 + 840 = 2520
What is the probability (if the roles are filled at random) of both Winston and Julia getting a part?
8 male for 2 spots
1 female for 1 spot
6 children for 2 spots
C₈,₂ * C₆,₂
C₈,₂ = = 8!/2!.6! = 28
C₆,₂ = 6!/2!.4! = 15
C₈,₂ * C₆,₂ = 28.15 = 420
p = 420/6300 = 0.067
![\bf ~~~~~~~~~~~~\textit{distance between 2 points}\\\\ \begin{array}{ccccccccc} &&x_1&&y_1&&x_2&&y_2\\ % (a,b) &&(~ -1 &,& -3~) % (c,d) &&(~ -7 &,& -5~) \end{array} \\\\\\ d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ r=\sqrt{[-7-(-1)]^2+[-5-(-3)]^2}\implies r=\sqrt{(-7+1)^2+(-5+3)^2} \\\\\\ r=\sqrt{36+4}\implies r=\sqrt{40}\\\\ -------------------------------](https://tex.z-dn.net/?f=%5Cbf%20~~~~~~~~~~~~%5Ctextit%7Bdistance%20between%202%20points%7D%5C%5C%5C%5C%0A%5Cbegin%7Barray%7D%7Bccccccccc%7D%0A%26%26x_1%26%26y_1%26%26x_2%26%26y_2%5C%5C%0A%25%20%20%28a%2Cb%29%0A%26%26%28~%20-1%20%26%2C%26%20-3~%29%20%0A%25%20%20%28c%2Cd%29%0A%26%26%28~%20-7%20%26%2C%26%20-5~%29%0A%5Cend%7Barray%7D%0A%5C%5C%5C%5C%5C%5C%0Ad%20%3D%20%5Csqrt%7B%28%20x_2-%20x_1%29%5E2%20%2B%20%28%20y_2-%20y_1%29%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0Ar%3D%5Csqrt%7B%5B-7-%28-1%29%5D%5E2%2B%5B-5-%28-3%29%5D%5E2%7D%5Cimplies%20r%3D%5Csqrt%7B%28-7%2B1%29%5E2%2B%28-5%2B3%29%5E2%7D%0A%5C%5C%5C%5C%5C%5C%0Ar%3D%5Csqrt%7B36%2B4%7D%5Cimplies%20r%3D%5Csqrt%7B40%7D%5C%5C%5C%5C%0A-------------------------------)
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