A hiring manager will conduct six interviews during one afternoon and

Question

A hiring manager will conduct six interviews during one afternoon and will also take two coffee breaks.

The eight time slots may be arranged in any order, subject to the following conditions:

• There must be at least two interviews scheduled between the two coffee breaks.
• Neither coffee break may be scheduled in the first or last time slot.

In how many different ways can the manager arrange the eight time slots?

A. 3 * 6!
B. 6 * 6!
C. 10 * 6!
D. 15 * 6!
E. 21 * 6!

gmatophobia · Answer

Let's assume there are  8  time slots as shown below -

s_1 \quad s_2 \quad s_3 \quad s_4 \quad s_5 \quad s_6 \quad s_7 \quad s_8

The coffee break cannot be set in either slot  s_1  or  s_8 , so one has to choose two slots between  s_2  and  s_7 , both inclusive.

Number of ways in which two slots can be selected from six available slots =  ^6C_2  ways = 15 ways

Among these 15 ways exists slots which are consecutive , such  s_2  and  s_3 ,  s_3  and  s_4  so on, and combinations, of breaks which are separated by only one interview, for example  s_2  and  s_4 ,  s_3  and  s_5  so on. We have to exclude them.

Number of consecutive slots = 5 ( s_2  &  s_3 ;  s_3  &  s_4 ;  s_4  &  s_5 ;  s_5  &  s_6 ;  s_6  &  s_7 )

Number of slots separated by only one interview slot = 4 ( s_2  &  s_4 ;  s_3  &  s_5 ;  s_4  &  s_6 ;  s_5  &  s_7 )

Remaining = 15 - 5 - 4 = 6

Therefore there are only 6 ways in which the coffee breaks can be arranged, and we will end up consuming two slots for that. The remaining 6 slots can be used for interview, and in these six slots the six interviews can be arranged in 6! ways.

Total = 6 * 6!

Option B

kevincan · Answer

Explained in great detail! Thanks for taking the time to write all this

Dereno · Answer

Let the eight time slots be P1, P2, P3, P4, P5, P6, P7, P8 respectively.

Coffee breaks cannot occupy position 1 and 8.

If coffee break is at position 2, then the next coffee break can be at position 5 or 6 or 7 = 3 ways.

Similarly, for coffee break position 3, the next coffee break can be either at position 6 or 7 = 2 ways.

For Coffee Break at position 4, the next coffee break can be position 7 = 1 way.

Total ways for coffee break = 6 ways.

For the positions 1 and 8, out of the 6 interview slots we select two = 6c2.

Arranging them = 2!

Remaining four interview slots for four interviews = 4c4 * 4!

So for the interview alone = 6c2 * 2! * 4! = 6!

For both interview and coffee break =  6! * 6  ways.

Option B

GraemeGmatPanda · Answer

We first choose two positions for the coffee breaks from the six available slots (positions 2 through 7), ignoring spacing conditions
C(6,2) =  \frac{6!}{2!4!}

We then subtract the invalid pairs where the breaks are fewer than two interviews apart (adjacent pairs and pairs two apart)
15 - (5 + 4) = 6

Next we arrange the six distinct interviews in the remaining six slots
6!

Multiplying the break arrangements by the interview arrangements gives the total
6  	imes  6!

Answer B

Hope this helps! 
ʕ•ᴥ•ʔ

A hiring manager will conduct six interviews during one afternoon and

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