Question

. Show all work. A disc jockey has 11 songs to play. Eight are slow songs, and three are fast songs. Each song is to be played only once. In how many ways can the disc jockey play the 11 songs if
• The songs can be played in any order.
• The first song must be a slow song and the last song must be a slow song.
• The first two songs must be fast songs.

Answer 1

hi polpak how r u today?

Answer 2

The number of ways you can permute k elements from a set of n elements is \[P(n,k) = \frac{n\!!}{(n-k)\!!}\]
\[\implies P(11,11) = \frac{11!}{(11-11)!} = \frac{11!}{0!} = 11!\]

Answer 3

I'm doing well!

Answer 4

there is only 11 songs played in order?

Answer 5

No. There's 11 factorial different ways the jocky can play 11 songs with no restriction on the order.

Answer 6

= 39916800

Answer 7

how did u get that number? 11*11=121

Answer 8

11*10*9*8*7*6*5*4*3*2*1 = 11! = 39916800

Answer 9

oh ok.. so u take each number 1-11 and multiply them..

Answer 10

Right. When he is choosing the first song there are 11 different ways he can pick it.
After that there are 10 songs he can choose from for the second song.
etc.

Answer 11

ok.. now what about the 2nd question..

Answer 12

Ok, well how many different songs can he pick from for the first song?

Answer 13

39916800

Answer 14

No, there aren't that many songs to choose from. Also he has to pick a slow song first.

Answer 15

so there is 11 songs total. and 8 r slow so it would be 8 songs to choose from that are slow

Answer 16

Ok, so there are 8 different choices for the first song.
How many options are there for the last song (after the first song gets picked)?

Answer 17

7 options

Answer 18

Ok, so he can pick the first and last songs 8*7 = 56 different ways.

Once he has done that he just has to figure out the order for the rest of the songs.. How many are there?

Answer 19

How many songs are left to pick from that is?

Answer 20

9 songs left.. after the 2 are taken out.. for being slow

Answer 21

or is it from the 56 ways?

Answer 22

No that's right. There are 9 songs left you have to pick an order for. So there is 9*8*...*2*1= 9! different ways to order those.

Therefore there is in total: 56*9! = 20,321,280 different ways you can play the songs with a slow at the beginning and a slow one at the end.

Answer 23

ok thats makes sense.

Answer 24

Now try the last one

Answer 25

there are 3 songs that are fast

Answer 26

and the first 2 have to be fast

Answer 27

2*1=3

Answer 28

2*1 = 2 actually.

Answer 29

oh yea.. duh lol

Answer 30

But the question is... When he goes to pick the first song, how many different ways can he do it?

Answer 31

2*11 ?

Answer 32

No. How many songs can he choose from for the first song?

Answer 33

3 songs to chose from

Answer 34

cuz they have to be fast songs

Answer 35

Right. And how many does he have to choose from after he picks the first one?

Answer 36

For the second song

Answer 37

2 songs

Answer 38

cuz the second one has to be fast also

Answer 39

And how many can he pick from for the 3rd song?

Answer 40

9 songs for the 3rd

Answer 41

And for the fourth?

Answer 42

8

Answer 43

etc

Answer 44

So the total number of different ways is:

Answer 45

(the product of all those options)

Answer 46

11

Answer 47

No...
3*2*9*8*7*6*...*2*1 =

Answer 48

18144 *2*1

Answer 49

36288

Answer 50

I think you're being confused when I say ... I mean the pattern continues..

\[\underbrace{3\cdot2}_{\text{First two songs}} \cdot \underbrace{9\cdot 8\cdot7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1}_{\text{all the rest}} = 22,177,280\]

Answer 51

oh ok..
so 3*2= 6* then 9*8*7*6*5*4*3*2*1=

Answer 52

take the 6 multiply times 9 8 7 6 5 4 3 2 1= my answer

Answer 53

Right

Answer 54

i have 2177280 for answer

Answer 55

i did it 3 times on caculator

Answer 56

That is what I got also.

Answer 57

so that would be my answer for the final question

Answer 58

Correct

Answer 59

ok now i understand this.. Thank you :)