Five A-list actresses are vying for the three leading roles

Five A-list actresses are vying for the three leading roles in the new film, "Catfight in Denmark." The actresses are Julia Robards, Merly Strep, Sally Fieldstone, Lauren Bake-all, and Hallie Strawberry. Assuming that no actresses has any advantage in getting any role, what is the probability that Julia and Hallie will star in the film together?

(A) 3/10
(B) 3/125
(C) 6/10
(D) 3/5
(E) 1/125

\(Probability=\frac{Number \ of \ favorable \ outcomes}{total \ Number \ of \ outcomes}\).

Now, total the total Number of outcomes will be \(C^3_5=10\), (number of ways to choose 3 different actresses out of 5 when order of selection doesn't matter);

As for the number of favorable outcomes: we want 2 places out of 3 to be taken by J and H: so {JH-?}, for the third spot we can choose ANY from the 3 actresses left M, S, L. So, there are 3 such favorable groups possible: {JH-M}, {JH-S}, {JH-L}. Or \(C^1_1*C^1_3\), where \(C^1_1=1\) is 1 way to choose J and H (as it's one group) and \(C^1_3=3\) is 3 ways to choose the third member;

Thus \(P=\frac{3}{10}\).

Answer: A.

Hope it helps.

Why isn't this approach right?

This is how I am doing it and I end up getting 2/5 which is not right.

I am doing the following:

Julia first (1/5) and Hallie second (1/4) = (1/5 * 1/4) OR
Hallie first (1/5) and Julia second (1/4) = (1/5 * 1/4) OR
Julia first (1/5) and Hallie third (1/3)=(1/5 * 1/3) OR
Hallie first (1/5) and Juila third (1/3)= (1/5 * 1/3) OR
Julia second (1/4) and Hallie third (1/3) = (1/4 * 1/3) OR
Hallie second (1/4) and Julia third (1/3) = (1/4 * 1/3)

I add them all up which totals to 2/5

Anything wrong here?

Here is your problem:
Julia first (1/5) and Hallie second (1/4) and anyone else third = (1/5 * 1/4 * 1) OR
Hallie first (1/5) and Julia second (1/4) and anyone else third = (1/5 * 1/4 * 1) OR
Julia first (1/5), someone other than Hallie second and Hallie third (1/3)=(1/5 * 3/4 * 1/3) OR
Hallie first (1/5), someone other than Julia second, and Juila third (1/3)= (1/5 * 3/4 * 1/3) OR
Someone other than Hallie and Julia first, Julia second (1/4) and Hallie third (1/3) = (3/5 * 1/4 * 1/3) OR
Someone other than Hallie and Julia first, Hallie second (1/4) and Julia third (1/3) = (3/5 * 1/4 * 1/3)

When you add them up, you get 6/20 = 3/10

Another way to do it:
Pick Hallie and Julia. Now you can pick the third actress in 3 ways so total number of ways of picking 3 actresses (including Hallie and Julia) = 3
Total number of ways of picking any 3 actresses out of 5 = 5*4*3/3! = 10
Probability that both Hallie and Julia will be picked = 3/10

I just count even though it may seem somehow basic.

Suppose we want to select 3 from 5 of following:

A-B-C-J-H

There are:
ABC
ABJ
ABH
ACJ
ACH
AJH
BCJ
BCH
BJH
CJH

The group which contain both J and H are 3

So the probability is: \(\frac{3}{10}\)

combinatorics tell us there are ten possibilities of combinations of actresses.

n!/k!(n-k)!

5 possibilities of choices - n = 5
3 will be seleced - k = 3

5*4*3*2*1/(3*2*1)(2*1) which reduces to (5*4)/(2*1) = 10.

There are 10 possible combinations you can choose to fill the 3 spots.

Now that you know there are 10, you need to find out which how many combinations you can make with the 2 selected actresses.

Julia and Halle can be grouped with any of the other 3 actresses, but there are no other possible combinations. So there are 3 groups that Julia and Halle could both be in.

3/10 is the answer.

Nice work by the creator of this question with the actress names. They all sound gorgeous.

Very simple solution, i would say is
Total number of ways in which you can select 3 out of 5 actresses is 5C3 = 10
Total number of ways in which you can select 2 actress for 3 slots is 3C2 = 3 (as all the actresses have same probability of getting selected)

Probability = 3/10.

Can someone please help me understand why didnt we use permutation here. Because
1. Julia, Hallie and Meryl
and
2. Hallie, Julia and Meryl

should be considered two different ways to select actresses, shouldnt it?

There are 3 different roles so selecting Julia, Hallie and Meryl is different from Hallie, Julia and Meryl. But we ignore the arrangements because we are looking for a probability.

P(Picking Hallie and Julia) = (No of ways in each Hallie and Julia will be picked)/(No of ways of picking any 3 actresses)
= (3*3!)/(10*3!)
Since we are calculating the probability, the arrangements of the numerator (3!) will get cancelled by the arrangements of the denominator (3!). So we can ignore them.

You can also reverse the question. If we'll hire 3 actresses, we'll fire 2 of them. We can focus on the two people we're firing. We just need to be sure we don't fire Julia or Hallie. There's a 3/5 chance the first person we fire is not J or H, and then a 2/4 chance the second person we fire is not J or H. So the answer is (3/5)(2/4) = 3/10.

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