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How many different ways can 2 students be seated in a row of

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ksharma12
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How many different ways can 2 students be seated in a row of [#permalink] New post  17 Feb 2010, 11:27
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How many different ways can 2 students be seated in a row of 4 desks, so that there is always at least one empty desk between the students?

(A) 2
(B) 3
(C) 4
(D) 6
(E) 12
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honeyrai
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Re: How many different ways can 2 students be seated in a row of [#permalink] New post  21 Feb 2010, 00:55
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ksharma12 wrote:
But that yields the same answer as 4C2? How do you solve use combinations or perm?

That comb above does not take into account the empty seat.

Here order is important, therefore, we'll use Perm.

All the possible ways in which 2 stud can be seated on 4 seats = 4P2.
Suppose, two students are always seated together. Then we'll treat them as 1 stud & the seats left to be filled are 3. Therefore no. of ways in which 2 stud are always seated together = 3P2.

Now, the no. of ways in which 2 students have a gap in between = total no. of ways - no. of ways in which students are seated together = 4P2-3P2 = 12-6 = 6
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LevFin7S
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Re: How many different ways can 2 students be seated in a row of [#permalink] New post  17 Feb 2010, 11:34
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3 ways to seat the students:
with two empty seats between
1 empty w/ one student on the left most
1 empty....right most

two students can be interchanged

3x2=6
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ksharma12
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Re: How many different ways can 2 students be seated in a row of [#permalink] New post  17 Feb 2010, 11:47
But that yields the same answer as 4C2? How do you solve use combinations or perm?

That comb above does not take into account the empty seat.
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Re: How many different ways can 2 students be seated in a row of [#permalink] New post  17 Feb 2010, 12:04
Take two cases.

case 1... 2 spaces in between that means C1 _ _ C2 or C2_ _ C1

case 2.... 1 space in between.
now consider c1_c2 as one element
Number of ways of arranging these in 4 seats = \(2p2*2!\) = 4

2p2 because we have considered c1_c2 as one element , 2! is rearrangement between c1 and c2

total = 6..whats OA?
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Re: How many different ways can 2 students be seated in a row of [#permalink] New post  24 Feb 2010, 01:18
ksharma12 wrote:
How many different ways can 2 students be seated
in a row of 4 desks, so that there is always at
least one empty desk between the students?

(A) 2
(B) 3
(C) 4
(D) 6
(E) 12
Easy Question I know. Explanation please?


Possible patterns = 1x2x, 1xx2, x1x2, 2x1x, x2x1, 2xx1 = 6 hence D

Whats the OA
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Re: How many different ways can 2 students be seated in a row of [#permalink] New post  13 Jul 2014, 01:31
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Total cases : 12 ( student one has 4 options and student two has three options, 4x3=12)
Non-favourable cases : 6 (when two students sit together. students in desk 1 and desk 2 , in desk 2 and desk 3, in desk 3 and desk 4) for each of these cases there are two possibilities because the positions can be interchanged. hence 2x3=6.

SO favourable cases : 12-6=6.
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How many different ways can 2 students be seated in a row of [#permalink] New post  26 Nov 2015, 10:49
It is important to see that the empty desks are not distinct (I missed this the first time, because I considered them to be)
We have "4" students (2 normal students {\(A\),\(B\)} and 2 students {\(E_1\),\(E_2\)} that we can consider empty desks) and 4 places.
Total = \(\frac{4!}{2!}\) since \(AE_1E_2B=AE_2E_1B\)
Cases in which the two students are together {{\(A\),\(B\)}, \(E_1\), \(E_2\)} = 3!*2!/2! since \(ABE_1E_2\) = \(ABE_2E_1\)
Total - together = 12 - 6 = 6
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Re: How many different ways can 2 students be seated in a row of [#permalink] New post  26 Nov 2015, 16:44
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Hi All,

Since the answers to this question are all small, and the question has a strong 'visual' component to it, we can 'brute force' the answer by drawing some pictures.

If we call the two students 'X' and 'Y', we have the following possibilities:

X _ Y _
Y _ X _

X _ _ Y
Y _ _ X

_ X _ Y
_ Y _ X

Total options: 6

Final Answer:
Show Spoiler
D


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Re: How many different ways can 2 students be seated in a row of [#permalink] New post  06 Aug 2021, 22:57
Students: A, B
Desks: _ _ _ _
Case 1: A _ B _ (A&B can switch places) : 2 Ways
Case 2: _ A _ B ( A&B can switch places) : 2 Ways
Case 3: A _ _ B (A &b B can switch places): 2 Ways

Total Ways: 2 + 2 + 2 = 6
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