How many different ways can a director select 4 actors from a gro...

Question

How many different ways can a director select 4 actors from a group of 20 actors to attend a workshop on performing in rock musicals?

Accepted Answer

Welcome back. I'm so glad you're here, john's got a vacation coming up. He wants to pack four shirts and he's got 11 to choose from. So how do we figure out how many different ways he can select those four shirts? The first question is, does order matter if he's got a dress shirt and a polo shirt? Does it matter if he puts the dress shirt into his suitcase before the polo shirt? It doesn't order. Doesn't matter. So we know we're dealing with combination. Combination notation tells us we've got a combination of end things taken R at a time And things. How many things does he have? He has 11 shirts total. And how many is he taking at a time from those 11? He's taking four at a time. So we've got 11 things taken four at a time. We recall from previous lessons that the formula for combination is N. Factorial over parentheses and minus R. And parentheses. Factorial times R. Factorial. And in this case now we can fill in the blanks in the numerator, N is 11 factorial, N -R. While that's 11 -4 factorial. R. Is for factorial. Great, let's simplify numerator is 11 factorial. 11 minus four is seven factorial and times four factorial. Okay, I am going to write out in long form what we're doing here and then show you how you can kind of skip this step in the future. We've got in the numerator 11 times 10 times nine times eight times seven times six times five times four times three times two times one. That's the factorial in the denominator we've got seven times six times five times four times three times two times one times four times three times two times one. And a lot of those cancel out starting here with that seven. So seven's cancel out sixes, fives fours threes twos and ones. Another way to think about this is we've got in the numerator the 11 factorial and in the denominator two different factorial. Which one's bigger, the seven. So we know we could write the numerator is 11 times 10 times nine times eight times seven factorial. That's seven times six times five times four etcetera. And the denominator could be seven factorial times four factorial. And those seven factorial can cancel out and that leaves us with 11 times 10 times nine times eight in the numerator. And if your calculator can do factorial, you can just type that in for the denominator. But if not you can type in four times three times two and you don't have to type times one. But I'm writing that so that you can see it's there. And the calculator gives us the answer of 330. That's a lot of choices for john to make before his vacation matches with answer choice B while done. We'll catch you on the next one

How many different ways can a director select 4 actors from a group of 20 actors to attend a workshop on performing in rock musicals?

Key Concepts

Combinations

Factorial

Binomial Coefficient

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