In how many ways can five girls stand in line if Maggie and Lisa cannot stand next to each other?
(A) 112
(B) 96
(C) 84
(D) 72
(E) 60
The OA is the option D.
What is the formula to solve this PS question? Combinations or Permutations? I am confused.
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In how many ways can five girls stand in line if . . . .
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Hi M7MBA,
We're asked for the number of ways that can five girls (including Maggie and Lisa) can stand in line IF Maggie and Lisa CANNOT stand next to each other.
To start, if there were no 'restrictions' on who could stand where, then there would be 5! = 120 possible arrangements. However, there IS a restriction, so we have to calculate the number of options that have to be removed from the 120...
If the first 2 people in line were Maggie and Lisa, then we would have....
M L (3)(2)(1) = 6 options that don't fit
L M (3)(2)(1) = 6 options that don't fit
12 total options that don't fit
If the 2nd and 3rd people in line were Maggie and Lisa, then we would have....
(3) M L (2)(1) = 6 options that don't fit
(3) L M (2)(1) = 6 options that don't fit
12 total options that don't fit
This pattern continues when they're the 3rd and 4th people and when they're the 4th and 5th people. Thus, there are 4(12) = 48 options that don't fit.
120 - 48 = 72 ways to arrange the 5 girls in line.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
We're asked for the number of ways that can five girls (including Maggie and Lisa) can stand in line IF Maggie and Lisa CANNOT stand next to each other.
To start, if there were no 'restrictions' on who could stand where, then there would be 5! = 120 possible arrangements. However, there IS a restriction, so we have to calculate the number of options that have to be removed from the 120...
If the first 2 people in line were Maggie and Lisa, then we would have....
M L (3)(2)(1) = 6 options that don't fit
L M (3)(2)(1) = 6 options that don't fit
12 total options that don't fit
If the 2nd and 3rd people in line were Maggie and Lisa, then we would have....
(3) M L (2)(1) = 6 options that don't fit
(3) L M (2)(1) = 6 options that don't fit
12 total options that don't fit
This pattern continues when they're the 3rd and 4th people and when they're the 4th and 5th people. Thus, there are 4(12) = 48 options that don't fit.
120 - 48 = 72 ways to arrange the 5 girls in line.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
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Good arrangements = total possible arrangements - bad arrangements.M7MBA wrote:In how many ways can five girls stand in line if Maggie and Lisa cannot stand next to each other?
(A) 112
(B) 96
(C) 84
(D) 72
(E) 60
Total arrangements:
Number of ways to arrange the 5 children = 5! = 120.
Bad arrangements:
In a bad arrangement, M and L stand next to each other.
To count the bad arrangements, put M and L in a BLOCK, as follows:
[ML].
Let the other 3 children be A, B and C.
Now count the number of ways to arrange the 4 elements [ML], A, B and C.
Number of ways to arrange the 4 elements [ML], A, B and C = 4! = 24.
Since [ML] can be reversed to [LM], the result above must be doubled:
2*24 = 48.
Good arrangements:
Total possible arrangements - bad arrangements = 120-48 = 72.
The correct answer is D.
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My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.
As a tutor, I don't simply teach you how I would approach problems.
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We can use the formula:M7MBA wrote:In how many ways can five girls stand in line if Maggie and Lisa cannot stand next to each other?
(A) 112
(B) 96
(C) 84
(D) 72
(E) 60
Number of ways Maggie is not next to Lisa = total number of arrangements - number of ways Maggie is next to Lisa
The total number of arrangements with no restrictions is 5! = 120.
Maggie is next to Lisa can be shown as:
[M-L] - A - B - C
Since Maggie and Lisa are now represented as one person, there are 4! ways to arrange the group and 2! ways to arrange Maggie and Lisa. Thus, we have 4! x 2! = 24 x 2 = 48 ways for Maggie to stand next to Lisa.
Thus, the number of ways to arrange Maggie and Lisa such that they are not together is 120 - 48 = 72.
Answer: D
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