2 teachers and 3 students line up in a row. How many different arrange

Question

2 teachers and 3 students line up in a row. How many different arrangements are possible if the teachers cannot be adjacent to each other?
 
A. 60 
B. 72 
C. 81 
D. 90 
E. 100

luffy_ueki · Accepted Answer

There are 2 teachers, 3 students => t1,t2,s1,s2,s3
_ _ _ _ _ (5 positions)

Total number of arrangements possible (without any condition) = 5! =120
Lets find the arrangement in which both teachers ARE together 
t1 t2 s1 s2 s3
lets group t1 & t2 as one entity (to ensure they are together during the arrangements) and arrange
=> (t1 t2) s1 s2 s3 
No=4! x 2! =48
2! because of t1,t2 & t2,t1.

Required number of arrangements = 5! - 4!*2! = 120 -48 = 72

chetan2u · Answer

Hi..

place 3 students in a row with gaps........ _S_S_S_
so there will be 4 gaps as shown above, which can be filled up by 2 teachers
so 4C2*2! = 12
the 3 students can be seated in 3!=6 ways

total 12*6=72

MathRevolution · Answer

=>

Since at least one student should be between the two teachers, we should consider complementary cases. This means we should calculate the difference between the total number of arrangements and the number of arrangements in which the teachers are adjacent to each other.

The total number of ways in which 5 people can stand in a row is 5!= 120.
The total number of arrangements with the teachers adjacent to each other is 4! * 2!.
Thus, the total number of permitted arrangements is 5! – 4! * 2! = 120 – 48 = 72.

Therefore, the answer is B.

Answer: B

syedazeem3 · Answer

I'm a bit confused - how did you figure out this step:"The total number of arrangements with the teachers adjacent to each other is 4! * 2!"?

Thanks,
syed

Dev23 · Answer

No of possible arrangements=4*3*3*2*1=72

Vegita · Answer

MathRevolution

How did you figure this out so quickly?

IanStewart · Answer

If the two teachers must sit together, you can think of them as if they were a single person. So then we have just 4 "people", and we can arrange 4 people in 4! ways. But we need to multiply that by 2!, because we could put the two teachers in 2! = 2 different orders.

So if you had a similar question, with 10 people, 3 of whom are teachers, then if the 3 teachers needed to sit together, you could first think of the 3 teachers as if they were one person, so we'd have 8 "people" in total (7 students, and 1 group of teachers), and we could arrange those "people" in 8! orders. But we would then need to multiply that by 3! because the three teachers could be in 3! different orders, so the answer would be 8! * 3!.

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2 teachers and 3 students line up in a row. How many different arrange

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2 teachers and 3 students line up in a row. How many different arrange

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