Question

Stats help
Q.From five boys & 4 girls a committe of 5 is to be formed.In how many ways can this be done to include atleast one girl.

Answer 1

I hope you remember what
\[\Large ^nC_r\] means...

Answer 2

It wasn't that...:D you thought me with different formula.

Answer 3

But i know it's like nCr =nPr/n

Answer 4

It's taught.
And yes, I suppose so.
Let's highlight the difference between \[\Large ^nP_r\]

and\[\Large ^nC_r\]

Answer 5

put numbers instead of N and R so i could actually tell you how..:D

Answer 6

\[\Large ^nP_r\]
counts how many ways you can arrange r elements in a row, from a set of n.


On the other hand
\[\Large ^nC_r\]
counts how many ways you can CHOOSE r elements from a set of n.


In other words, for \(\Large ^nP_r\), order matters, while in \(\Large ^nC_r\), order does not matter.

Answer 7

Now in forming a committee, you don't exactly arrange the members in a row, and it isn't especially important how you arrange them, so you can bet that in questions such as "how many ways can a committee blah blah blah blah" that the key thing to remember is \(\Large ^nC_r\)

Answer 8

All right xD

Answer 9

So, how do you plan on going about this question?
The additional complication is the fact that there must be at least one girl on the committee...

Answer 10

4C1 and 5C1?

Answer 11

4C1 = 4
5C1 = 5
That is trivial, and those don't really help you :)

It turns out, it is incredibly difficult to directly count the number of ways you can form a committee of five, while making sure to include at least one girl :)

Answer 12

Another way involves just counting the number of ways you can form a committee of 5 from these 9 choices... and that is just...?

Answer 13

How many ways can you form a committee of 5 from a set of 9 members?

Answer 14

(9 of course, since there are four girls and five boys)

Answer 15

Hm so 9C5?

Answer 16

That's right, so evaluate that, you get?

Answer 17

NOTE: That is not the answer yet...

Answer 18

It's not 9C5 - 1?

Answer 19

@KevinOrr I was getting there :P

Answer 20

Oh sorrz

Answer 21

But that's 9P5 :P

Answer 22

9C5 =9*8*7*6*5/5 right?

Answer 23

Nope...
\[\Large ^nC_r =\frac{\Large ^nP_r}{\color{red}{r!}} \]

Answer 24

So,9C5=9P5/5! =9*8*7*6*5/5*4*3*2*1

Answer 25

Yes... just simplify that nasty looking expression... first... before I tell you the rest :D

Answer 26

15,120/120=126.

Answer 27

slow down.
We get 9C5
What do you want 9C4 for?

Answer 28

And of course they'd be equal :/
\[\Large ^9C_5 = \frac{9!}{(9-5)!5!}=\frac{9!}{4!5!}=\frac{9!}{(9-4!)4!}=^9C_4\]

Answer 29

Now what after that? :D

Answer 30

Anyway, we have 126 ways to form a committee of 5, from these 9 members, right?

Answer 31

Yess.

Answer 32

But these also include those ways that don't include a single girl. In other words, ways that the question doesn't 'want' lol

Answer 33

Okay i get it.. :3 then?

Answer 34

Then... we take a way the number of ways you can form a committee of 5 WITHOUT including a single girl.

How do we do that?

How many ways can we form a committee of 5 without including girls?

Answer 35

We have to include at least one girl the questions says,No?

Answer 36

Yes. So from the total number of ways to form a committee of 5 out of the 9 members, we HAVE TO take away the number of ways to form a committee of 5 WITHOUT GIRLS.

Answer 37

So how do you count the number of ways you can form a committee of 5 without girls?

Answer 38

9P5?

Answer 39

That's what we just did...
9P5 counts the number of ways you can form a committee if you choose from all 9 of them, but remember, no girls?

How many of them are not girls?

Answer 40

5.

Answer 41

So, to form a committee of 5 without girls, we may only choose from those 5 boys...
So that means?

Answer 42

5P4?

Answer 43

4?
Why not 5?
You're forming a committee of 5, not 4 -.-

Answer 44

Yeah that's what I'm thinking won't it be 5P5?

Answer 45

wait, no, 5C5

Answer 46

Why did you use P all of a sudden? -.-

Answer 47

How do i know when to use P or C? :o

Answer 48

Wasn't that kind of the first thing I said? -.-

Answer 49

Anyway,5C5=5*4*3*2*1/5*4*3*2*1 =120/120 =1.

Answer 50

You use P when order matters.
You uce C when it doesn't.

And in forming committees, order doesn't matter.

Answer 51

Okay, so 5C5 is 1
Means there is only one way to form a committee of 5 without any girls.

Answer 52

You take that away from the TOTAL number of ways you can form a committee of 5 from the 9,and that's your answer.

Answer 53

Sorry but i have to say this *how do we know when order matters or not?* sorry...Sir.

Answer 54

Use your logic.
But look for keywords... if you see the word 'arrange' or anything similar, then order matters.

Answer 55

on the other hand, you see the word 'form' or 'choose' (though this is ambiguous, look out for 'arrange') then order probably doesn't matter

Answer 56

Ah i see.. :3 SO the answer is 125 ways.

Answer 57

Yes.

Answer 58

Am i not your favourite student? :3

Answer 59

That remains to be seen :|

Answer 60

Seen,ah?

Answer 61

Vamos a ver.

Answer 62

Bien entonces ... Pero cuando?

Answer 63

Si no sabes ahora, nunca sabrás.

Answer 64

Eso es tocar .. pero no lo sé: P

Answer 65

Pero tu sabes que yo tengo muchos estudiantes que he ayudado :)

Answer 66

Tu eres unica una :3

Answer 67

Das besondere? : D

Answer 68

Wir werden sehen. Was denkst du?

Answer 69

Ich denke, was Sie denken, und Sie wissen, was Sie denken, nein? : D

Answer 70

Nein, ich glaube dass du kannst nicht immer denken was ich denke.

Answer 71

Autsch ...

Answer 72

Wenn du glaubst dass wir sind Freunde, sag doch 'du' und nicht 'Sie'

Wir brauchen nicht sehr formal sein :)

Answer 73

Acabo de copiar lo que dice la traducción :)