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
Whole numbers<span><span>\greenD{\text{Whole numbers}}Whole numbers</span>start color greenD, W, h, o, l, e, space, n, u, m, b, e, r, s, end color greenD</span> are numbers that do not need to be represented with a fraction or decimal. Also, whole numbers cannot be negative. In other words, whole numbers are the counting numbers and zero.Examples of whole numbers:<span><span>4, 952, 0, 73<span>4,952,0,73</span></span>4, comma, 952, comma, 0, comma, 73</span>Integers<span><span>\blueD{\text{Integers}}Integers</span>start color blueD, I, n, t, e, g, e, r, s, end color blueD</span> are whole numbers and their opposites. Therefore, integers can be negative.Examples of integers:<span><span>12, 0, -9, -810<span>12,0,−9,−810</span></span>12, comma, 0, comma, minus, 9, comma, minus, 810</span>Rational numbers<span><span>\purpleD{\text{Rational numbers}}Rational numbers</span>start color purpleD, R, a, t, i, o, n, a, l, space, n, u, m, b, e, r, s, end color purpleD</span> are numbers that can be expressed as a fraction of two integers.Examples of rational numbers:<span><span>44, 0.\overline{12}, -\dfrac{18}5,\sqrt{36}<span>44,0.<span><span> <span>12</span></span> <span> </span></span>,−<span><span> 5</span> <span> <span>18</span></span><span> </span></span>,<span>√<span><span> <span>36</span></span> <span> </span></span></span></span></span>44, comma, 0, point, start overline, 12, end overline, comma, minus, start fraction, 18, divided by, 5, end fraction, comma, square root of, 36, end square root</span>Irrational numbers<span><span>\maroonD{\text{Irrational numbers}}Irrational numbers</span>start color maroonD, I, r, r, a, t, i, o, n, a, l, space, n, u, m, b, e, r, s, end color maroonD</span> are numbers that cannot be expressed as a fraction of two integers.Examples of irrational numbers:<span><span>-4\pi, \sqrt{3}<span>−4π,<span>√<span><span> 3</span> <span> </span></span></span></span></span>minus, 4, pi, comma, square root of, 3, end square root</span>How are the types of number related?The following diagram shows that all whole numbers are integers, and all integers are rational numbers. Numbers that are not rational are called irrational.
