Question

Calculate the number of ways that 3 or 4 prizes can be awarded to a group of 5 people if:

a) a member of the group is allowed to receive more than one prize

b) a member of the group cannot receive more than one prize.

Answer 1

May be begin with this now??

Answer 2

*we..

Answer 3

yes, sure.

Answer 4

Firstly for A part..

Answer 5

There are two conditions which says prizes can be 3 OR 4..

Right???

Answer 6

yes.

Answer 7

First let we have 3 prizes and 5 people...

See, our very first a) part says that any member can be given any number of prizes..

Right???

Answer 8

yep.

Answer 9

So, suppose you are distributing 3 prizes..

There are 5 people..

you will give first prize in how many ways??

You can give it to first person or second or third like that..

First prize you can give in how many ways??

Answer 10

5?

Answer 11

How can JayDS be wrong??

Yes you are right..

Now note that carefully: any people can have any number of prizes..

So, suppose you at first gave me the the first prize..

Now second prize you can give in how many ways??

NOTE: You can still given me a Prize..

Getting??

Tell me the number of ways for second prize?

Answer 12

lol, ur funny.
uh so lets see, if I gave u the first prize then for the second prize there should be 5 ways.

Answer 13

since I can still give u the second prize.

Answer 14

Yes you are right..

Like wise third prize can be given in how many ways and I can still receive a prize..

Answer 15

same, 5 ways.

Answer 16

Yes..

So you can distribute 3 prizes to 5 people in \(5 \times 5 \times 5\) ways..

Right??

Answer 17

yep.

Answer 18

So, now if I assume there are 4 prizes instead of 3 then 4 prizes you can give in total how many ways??

Same procedure..

Answer 19

yep, so 4x4x4 ways?

Answer 20

oh wait.

Answer 21

No...

Answer 22

exactly the same?

Answer 23

5x5x5 ways?

Answer 24

There are 4 prizes and 5 people..

just close..

Answer 25

See the ways you said are for 3 prizes where is the fourth prize gone??

Answer 26

yeh, I don't know lol.

Answer 27

I get confused easily, sorry.

Answer 28

First prize you can give in 5 ways,

Second prize you can give in 5 ways,

Third prize you can give in 5 ways,

Fourth prize you can give in 5 ways..

Now tell me what are the total number of ways??

Answer 29

20 ways

Answer 30

You have to multiply here..
I have multiplied it in case of 3 prizes 5*5*5...


So now tell me??

Answer 31

I think you are not getting what I said.

Answer 32

in the case of 4 prizes it will still be 5*5*5=125 ways right?
so the total would be 125+125 = 250 ways?

Answer 33

No..

See,

We have separately calculated for 3 prizes that is 5*5*5


Now we are calculating it for 4 prizes..

You have to just multiply them..

Aren't the number of ways to give 4 prizes 5*5*5*5 ??

Answer 34

ok so u just times 5^4 but why do we do this?

Answer 35

What??

Why we do multiplication instead of adding??

Answer 36

um, why do we multiply 5x5x5x5? I know it's to find the number of ways to give 4 prizes but I don't quite understand it.

Answer 37

See,

we find it separately..

First prize can be given in 5 ways..

Second in 5 third in 5 and finally fourth in 5..

Right??

Answer 38

yep.

Answer 39

Now do you know what Counting Principle says??

Answer 40

nope.

Answer 41

Counting Principle says that:

If there are two jobs such that one of them can be done in \(m\) ways and other can be done in \(n\) ways then, two jobs together can be completed in \(m \times n \) ways..

Answer 42

ok.

Answer 43

Similarly, here:

If first prize can be distributed in 5 ways, Second prize can be in 5 ways, Third can be in 5 ways and Fourth can be in 5 ways,

Then the four prizes in succession can be given in \(5 \times 5 \times 5\times5\) ways..


Not is it clear to you or not?

Answer 44

*Now..

Answer 45

ok, I believe I understand it now.

Answer 46

So tell me after multiplying what you get in both the cases??

When you distribute 3 prizes how many ways you get??

When you gave 4 then how many ways you get..??

Answer 47

5x5x5=125
5x5x5x5=625
125+625=750 ways

Answer 48

Yes now you are right..

Answer 49

If you are clear with this can we proceed for B??

Now we have to do the same but little bit different..

Answer 50

yep, sure.

Answer 51

I believe we will have to use factorial here in part b.

Answer 52

B> part says that one cannot have prize more than one..

Right??

Answer 53

yep.

Answer 54

No need to use factorial, Simple multiplication will serve the purpose..

Answer 55

kk.

Answer 56

First is the case with having 3 prizes..

Out of these 3, first prize you can give in how many ways??

Answer 57

three

Answer 58

No dear..

There are 5 people in total..

you can give it to first, second, third fourth or fifth..

So, how many ways??

Answer 59

oh yeh ==, 5.

Answer 60

Suppose you gave me the first prize..

You cannot give me the second prize now..

So second prize can be given in how many ways??

Answer 61

4.

Answer 62

Yes, now the last prize you can given in how many ways??

Answer 63

3

Answer 64

Yes good..

Now apply the counting principle here..

Total how many ways you get?

Answer 65

5x4x3=60 ways.

Answer 66

and so for the 4 prizes it will be 5x4x3x2=120 then?

Answer 67

You have really saved me..
Thanks..

Yes you are absolutely correct.. \(\large \color{green}\checkmark\)

Answer 68

So total how many ways for second (B) part??

Answer 69

180

Answer 70

Any doubt, anywhere??

Answer 71

lol, I'm really sorry! I am in fact a slow learner.

Answer 72

If you are a slow leaner then it very good..

Always in a race of turtle and rabbit, turtle wins..

Answer 73

yep, my only doubt is that I do understand it now, but I might forget how to do it properly in the test or other similar questions haha.

Answer 74

and thanks for the positive comment, may I ask what is your name by the way? I would like to address you properly.

Answer 75

I say honestly I am not good at Permutations And Combinations that is why I am saying that you should first listen to the lecture of your teacher..

He can explain you far better than me..

Answer 76

You can call me Water..

My real name is Gurpreet...

Answer 77

Yahoo is calling me I have to go now to help him..
Take care JayDS..

Answer 78

yep, I understand. and I am actually still in high school so I just have classes not lectures yet and I find that you explain quite well considering it's text that I am reading and not like me listening in class.