anonymous
  • anonymous
In how many ways can the five letters J, K, L, M and N be arranged such that L is not in the middle?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
i dont understand why it cant be 5 x 4 x 4 x 3 x 2
sdfgsdfgs
  • sdfgsdfgs
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?
anonymous
  • anonymous
5!

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sdfgsdfgs
  • sdfgsdfgs
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.
anonymous
  • anonymous
wait it's not supposed to be in the middle
sdfgsdfgs
  • sdfgsdfgs
hehee go along w me there plz?
anonymous
  • anonymous
So theres 4 ways?
sdfgsdfgs
  • sdfgsdfgs
missing a symbol after the 4 there...
anonymous
  • anonymous
4!
sdfgsdfgs
  • sdfgsdfgs
Now what happens when u subtract this from the results u got before with 5 letters and 5 positions?
anonymous
  • anonymous
huh?
Plasmataco
  • Plasmataco
ok.
Plasmataco
  • Plasmataco
how many positions can L BE IN the middle?
anonymous
  • anonymous
0
anonymous
  • anonymous
without = 4
Plasmataco
  • Plasmataco
...
Plasmataco
  • Plasmataco
In JKLMN, L is in the middle.
sdfgsdfgs
  • sdfgsdfgs
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
anonymous
  • anonymous
hmmmm
anonymous
  • anonymous
sorry im really not gettin it
sdfgsdfgs
  • sdfgsdfgs
@yomamabf think about it...or u can go along w the other way @Plasmataco is going :)
anonymous
  • anonymous
wait i dont get why u would add L in the middle?
anonymous
  • anonymous
when it says not to
sdfgsdfgs
  • sdfgsdfgs
so that those cases (4!) can be SUBTRACTED from the others (5!)
anonymous
  • anonymous
why do u subtract it
sdfgsdfgs
  • sdfgsdfgs
cuz' u are trying to find ways that L is NOT in the middle ;)
anonymous
  • anonymous
wouldnt it just be 5 x 4 x 4 x 3 x 2?
sdfgsdfgs
  • sdfgsdfgs
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)
anonymous
  • anonymous
so subtract it huh
sdfgsdfgs
  • sdfgsdfgs
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.
anonymous
  • anonymous
how do u do that method?
sdfgsdfgs
  • sdfgsdfgs
hahahaaa its slightly more complicated n dats why i did it my way...perhaps u can ask @Plasmataco to come back n show u? :)
sdfgsdfgs
  • sdfgsdfgs
OK.... for L not in the middle, it will be: L _ _ _ _ or _ L _ _ _ or _ _ _ L _ or _ _ _ _ L so far so good?
anonymous
  • anonymous
yes
ybarrap
  • ybarrap
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)
sdfgsdfgs
  • sdfgsdfgs
OK now consider the first case L _ _ _ _ how ways are there to fill KJMN to the 4 spaces?
anonymous
  • anonymous
i get it now thanks
sdfgsdfgs
  • sdfgsdfgs
ok :)
ybarrap
  • ybarrap
You're welcome
anonymous
  • anonymous
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
sdfgsdfgs
  • sdfgsdfgs
@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!

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