VaibNop wrote:
gracie wrote:
amod243 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
M & L can be separated in 6 ways, in these positions:
1 & 3
1 & 4
1 & 5
2 & 4
2 & 5
3 & 5
each of these 6 ways allows 12 possibilities:
2 for M & L:
ML
LM
* 6 for the other 3 girls:
ABC
ACB
BAC
BCA
CAB
CBA
6*12=72
D
Hi Friends,
Can anybody tell me how to solve this problem I am just bit modify this problem! In how many ways can five girls stand in line if Maggie, Lisa, and Jane cannot stand next to each other?Hello
VaibNop , I have looked at your attempt. The same is ok to visualize but time consuming for exam.
This is a problem of
combinatorics. Lets understand when only 2 of the 5 can not stand side by side . then we will apply the logic in your query 3 of 5.
Case 01: when only 2 of the 5 can not stand side by sideNow the total number of ways 5 people can be arranged
=5P5=5!=120The Q asked in how many cases 2 can not stand side by side.
Lets find the #cases where 2 can stand side by side . ( then, # ways when 2 donot stand side by side = total # ways 5 people stand - #cases where 2 can stand side by side)
Lets assume these 2 are actually 1 object , hence our modified total = 4 objects. Now the total number of ways 4 objects can be arranged
=4P4=4!=24.Now the 2 objects , which we considered to be a single object can be arranged among them selves in ways
=2P2=2!=2 .#cases where 2 can stand side by side =
24*2 = 48Thus ,
# ways when 2 donot stand side by side = total # ways 5 people stand - #cases where 2 can stand side by side = 120 - 48 = 72 waysCase 02: when only 3 of the 5 can not stand side by side ....................Please note how we are just plugging in the values to our earlier understanding.
Now the total number of ways 5 people can be arranged
=5P5=5!=120The Q asked in how many cases 3 can not stand side by side.Lets assume these 3 are actually 1 object , hence our modified total = 3 objects. Now the total number of ways 3 objects can be arranged
=3P3=3!=6.Now these 3 objects , which we considered to be a single object can be arranged among them selves in ways
=3P3=3!=6 .#cases where 3 can stand side by side =
6*6 = 36Thus ,
# ways when 3 donot stand side by side = total # ways 5 people stand - #cases where 3 can stand side by side = 120 - 36 = 84 ways _________________
Please let me know if I am going in wrong direction.
Thanks in appreciation.