In how many ways can a teacher in a kindergarten school

Question

In how many ways can a teacher in a kindergarten school arrange a group of 6 children (Susan, Karen, lei, Tim, joy and Zen) on 6 identical chairs in a straight line so that Susan is on the left of Tim?

A. 720
B. 360
C. 240
D. 120
E. 60

A. 720 
B. 360
C. 240 
D. 120
E. 60

GMATinsight · Accepted Answer

Method-1:

Total ways in which 6 children can be arranged on 6 chairs = 6*5*4*3*2*1 = 6! = 720

But in half cases Susan will be left of Tim and in other half of cases Tim will be on left of Susan

i.e. Desired cases in which Susan is on the left of Tim = (1/2)*720 = 360

Method-2:

We have six vacant chairs _ _ _ _ _ _

out of which 2 have to be occupied by Tim and Susan, No. of ways of selecting chairs for Tim and Susan = 6C2 = 15

No. of ways of arranging Tim an Susan on selected chairs = 1 (Because Susan must be on the left of Tim)

Total ways of arranging 4 remaining children on 4 remaining chairs = 4*3*2*1 = 4! = 24

i.e. Total Arrangement = 6C2 * 1 * 4! = 15*1*24 = 360

Method-3:

Case 1: S _ _ _ _ _ 
Total ways of arranging Tim to the right of Susan = 5
Total ways of arranging Remaining 4 children = 4!
i.e. Total Arrangement in this case = 5*4!

Case 2: _ S _ _ _ _ 
Total ways of arranging Tim to the right of Susan = 4
Total ways of arranging Remaining 4 children = 4!
i.e. Total Arrangement in this case = 4*4!

Case 3: _ _ S _ _ _ 
Total ways of arranging Tim to the right of Susan = 3
Total ways of arranging Remaining 4 children = 4!
i.e. Total Arrangement in this case = 3*4!

Case 4: _ _ _ S _ _ 
Total ways of arranging Tim to the right of Susan = 2
Total ways of arranging Remaining 4 children = 4!
i.e. Total Arrangement in this case = 2*4!

Case 5: _ _ _ _ S _ 
Total ways of arranging Tim to the right of Susan = 1
Total ways of arranging Remaining 4 children = 4!
i.e. Total Arrangement in this case = 1*4!

Total Arrangements = 5*4!+5*4!+4*4!+3*4!+2*4!+1*4! = 4!(5+4+3+2+1) = 24*15 = 360

Answer: Option B

Bunuel · Answer

6 children can be arranged in 6! ways. Now, in half of these arrangements Susan will be on the left of Tim and in half she'l be on the right of Tim, so the answer is 6!/2=360.

Answer: B.

Similar questions to practice:
https://gmatclub.com/forum/susan-john-d ... 30743.html
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harshavmrg · Answer

Hi Bunuel,

can you please help me understand how did u come to the conclusion that for half the arrangements S is to the left of T and other half to the right of T?

Regards
Harsha

Bunuel · Answer

S and T can have ONLY TWO positions: S to the right of T OR to the left, how else? Now, why should S be in more cases to the right of T than to the left? Does probability favors either of them? No. Hence exactly in half of these cases S will be to the right of T and in half of these cases to the left. Check these links for more: susan-john-daisy-tim-matt-and-kim-need-to-be-seated-in-130743.html goldenrod-and-no-hope-are-in-a-horse-race-with-6-contestants-82214.html in-how-many-different-ways-can-the-letters-a-a-b-91460.html

JeffTargetTestPrep · Answer

We see that the 6 students can be arranged in 6! = 720 ways. We also can determine that the number of ways in which Susan can be placed to the left of Tim is equal to the number of ways in which she can be placed to the right of Tim. Half the time she is to his right and half the time she is to his left. Thus, the number of ways the teacher can arrange the group with Susan to the left of Tim is 720/2 = 360 ways.

Answer: B

plalud · Answer

Note that if we read the question as number of ways where the two children are right next to one another, then the answer becomes 5!. Indeed assuming Tim is the first person to sit, he only has 5 spots, not 6, to choose from - he cannot sit at the far left.
So answer is (6-1)!

akadiyan · Answer

In how many ways can a teacher in a kindergarten school arrange a group of 6 children (Susan, Karen, lei, Tim, joy and Zen) on 6 identical chairs in a straight line so that Susan is on the left of Tim?

A. 720
B. 360
C. 240
D. 120
E. 60

Total number of possibilities= 6! = 720

constraints:
We need to arrange 2 chairs for susan and tim so that susan is on the left of tim

Step 1:
Possibility of arranging 2 chairs for Susan and Tim = The possibilities is 6C2 = \frac{6!}{2!*4} ! = 15

Step 2:
Number of arrangement for other 4 children = 4! = 24

Total number of arrangements = 15 * 24 = 360

Ans:  B

kaustubhkohli12 · Answer

Hi , Can you please explain why my logic is not correct here ? I thought of ST as one pair and the remaining are 4 . So in turns 5 numbers . So 5! = 120 . Can you explain what i missed here ?

Bunuel · Answer

Your method treats  Susan and Tim as one pair , so it counts only cases where they sit  next to each other .

But the question only says Susan is  to the left of  Tim, not necessarily immediately next to Tim.

So we count all 6! arrangements and take half of them:

6!/2 = 360

Answer: B.

In how many ways can a teacher in a kindergarten school

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In how many ways can a teacher in a kindergarten school

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