Question

In how many ways can the five letters J, K, L, M and N be arranged such that L is not in the middle?

Answer 1

i dont understand why it cant be 5 x 4 x 4 x 3 x 2

Answer 2

first u need to figure out: how many ways can 5 letters be arranged (including the cases where L is in the middle). what is that equal to?

Answer 3

5!

Answer 4

Good job! :)

Now consider this: how many ways can 4 letters, JKMN, be arranged in 4 positions? There are 4 open positions only because this assumes L is in the middle.

Answer 5

wait it's not supposed to be in the middle

Answer 6

hehee go along w me there plz?

Answer 7

So theres 4 ways?

Answer 8

missing a symbol after the 4 there...

Answer 9

4!

Answer 10

Now what happens when u subtract this from the results u got before with 5 letters and 5 positions?

Answer 11

huh?

Answer 12

ok.

Answer 13

how many positions can L BE IN the middle?

Answer 14

0

Answer 15

without = 4

Answer 16

...

Answer 17

In JKLMN, L is in the middle.

Answer 18

ways 5 letters be arranged (including the cases where L is in the middle) = 5!
ways 4 letters, JKMN, be arranged in 4 positions w/ L in the middle = 4!

so 5!-4! = ways five letters J, K, L, M and N be arranged such that L is not in the middle

Answer 19

hmmmm

Answer 20

sorry im really not gettin it

Answer 21

@yomamabf think about it...or u can go along w the other way @Plasmataco is going :)

Answer 22

wait i dont get why u would add L in the middle?

Answer 23

when it says not to

Answer 24

so that those cases (4!) can be SUBTRACTED from the others (5!)

Answer 25

why do u subtract it

Answer 26

cuz' u are trying to find ways that L is NOT in the middle ;)

Answer 27

wouldnt it just be 5 x 4 x 4 x 3 x 2?

Answer 28

total cases = (cases L in middle) + (cases L not in the middle)

n ur prob is asking u to find (cases L not in the middle)

Answer 29

so subtract it huh

Answer 30

Correct - the other way is to directly calculate (cases L not in the middle) n i think that was where the other helper was going.

Answer 31

how do u do that method?

Answer 32

hahahaaa its slightly more complicated n dats why i did it my way...perhaps u can ask @Plasmataco to come back n show u? :)

Answer 33

OK....

for L not in the middle, it will be:

L _ _ _ _ or
_ L _ _ _ or
_ _ _ L _ or
_ _ _ _ L

so far so good?

Answer 34

yes

Answer 35

Here are all the ways to permute the 4 letters

{J, K, M, N} | {J, K, N, M} | {J, M, K, N} | {J, M, N, K} | {J, N, K, M} | {J, N, M, K} | {K, J, M, N} | {K, J, N, M} | {K, M, J, N} | {K, M, N, J} | {K, N, J, M} | {K, N, M, J} | {M, J, K, N} | {M, J, N, K} | {M, K, J, N} | {M, K, N, J} | {M, N, J, K} | {M, N, K, J} | {N, J, K, M} | {N, J, M, K} | {N, K, J, M} | {N, K, M, J} | {N, M, J, K} | {N, M, K, J} (total: 4!=24)

4! places to permute J,K,M,N, while keeping L in the middle is the same as permuting just J,K,M,N because L has no freedom to permute to another position.

There are 5! ways to permute all the letters (see below)

Here are all the ways to permute the 5 letters. Count the ones with L in the Middle, there will be 24. The remaining will be those permutations without L in the middle. That's why you subtract: 5! - 4!. It's easier to count those without L in the middle then subtract than to directly count those with L in the middle.

{J, K, L, M, N} | {J, K, L, N, M} | {J, K, M, L, N} | {J, K, M, N, L} | {J, K, N, L, M} | {J, K, N, M, L} | {J, L, K, M, N} | {J, L, K, N, M} | {J, L, M, K, N} | {J, L, M, N, K} | {J, L, N, K, M} | {J, L, N, M, K} | {J, M, K, L, N} | {J, M, K, N, L} | {J, M, L, K, N} | {J, M, L, N, K} | {J, M, N, K, L} | {J, M, N, L, K} | {J, N, K, L, M} | {J, N, K, M, L} | {J, N, L, K, M} | {J, N, L, M, K} | {J, N, M, K, L} | {J, N, M, L, K} | {K, J, L, M, N} | {K, J, L, N, M} | {K, J, M, L, N} | {K, J, M, N, L} | {K, J, N, L, M} | {K, J, N, M, L} | {K, L, J, M, N} | {K, L, J, N, M} | {K, L, M, J, N} | {K, L, M, N, J} | {K, L, N, J, M} | {K, L, N, M, J} | {K, M, J, L, N} | {K, M, J, N, L} | {K, M, L, J, N} | {K, M, L, N, J} | {K, M, N, J, L} | {K, M, N, L, J} | {K, N, J, L, M} | {K, N, J, M, L} | {K, N, L, J, M} | {K, N, L, M, J} | {K, N, M, J, L} | {K, N, M, L, J} | {L, J, K, M, N} | {L, J, K, N, M} | {L, J, M, K, N} | {L, J, M, N, K} | {L, J, N, K, M} | {L, J, N, M, K} | {L, K, J, M, N} | {L, K, J, N, M} | {L, K, M, J, N} | {L, K, M, N, J} | {L, K, N, J, M} | {L, K, N, M, J} | {L, M, J, K, N} | {L, M, J, N, K} | {L, M, K, J, N} | {L, M, K, N, J} | {L, M, N, J, K} | {L, M, N, K, J} | {L, N, J, K, M} | {L, N, J, M, K} | {L, N, K, J, M} | {L, N, K, M, J} | {L, N, M, J, K} | {L, N, M, K, J} | {M, J, K, L, N} | {M, J, K, N, L} | {M, J, L, K, N} | {M, J, L, N, K} | {M, J, N, K, L} | {M, J, N, L, K} | {M, K, J, L, N} | {M, K, J, N, L} | {M, K, L, J, N} | {M, K, L, N, J} | {M, K, N, J, L} | {M, K, N, L, J} | {M, L, J, K, N} | {M, L, J, N, K} | {M, L, K, J, N} | {M, L, K, N, J} | {M, L, N, J, K} | {M, L, N, K, J} | {M, N, J, K, L} | {M, N, J, L, K} | {M, N, K, J, L} | {M, N, K, L, J} | {M, N, L, J, K} | {M, N, L, K, J} | {N, J, K, L, M} | {N, J, K, M, L} | {N, J, L, K, M} | {N, J, L, M, K} | {N, J, M, K, L} | {N, J, M, L, K} | {N, K, J, L, M} | {N, K, J, M, L} | {N, K, L, J, M} | {N, K, L, M, J} | {N, K, M, J, L} | {N, K, M, L, J} | {N, L, J, K, M} | {N, L, J, M, K} | {N, L, K, J, M} | {N, L, K, M, J} | {N, L, M, J, K} | {N, L, M, K, J} | {N, M, J, K, L} | {N, M, J, L, K} | {N, M, K, J, L} | {N, M, K, L, J} | {N, M, L, J, K} | {N, M, L, K, J} (total: 5!=120)

Answer 36

OK now consider the first case L _ _ _ _
how ways are there to fill KJMN to the 4 spaces?

Answer 37

i get it now thanks

Answer 38

ok :)

Answer 39

You're welcome

Answer 40

4 x 3 x 4 x 2 x 1 = 96 You use the 4 in the beginning because the middle letter is already used by a letter other than L so it cant be a 5 in the beginning

Answer 41

@yomamabf correct - then if u consider there are 4 ways:

L _ _ _ _ or
_ L _ _ _ or
_ _ _ L _ or
_ _ _ _ L

So the ans becomes 4*4!
which is equal to 5!-4!