Three friends all in a row

Question

Six people, including three friends Ann, Ben, and Cy, are seated uniformly at random in a row of $6$ seats.

What is the probability that the three friends occupy three consecutive seats (in any order among themselves)?

Accepted Answer

The clean tool for "all together" is the block trick: treat Ann, Ben, and Cy as one glued unit. That leaves 4 units to arrange (the block plus the 3 others), which can be ordered in 4! ways, and inside the block the three friends can be arranged in 3! ways. Total favorable arrangements: 4! 	imes 3! = 24 	imes 6 = 144. All arrangements of 6 people number 6! = 720, so P = \frac{144}{720} = \frac{1}{5}. A quick cross-check on the block count: the block of 3 can start at seat 1, 2, 3, or 4, that is 4 positions, matching the 4! ways of placing the block among the units. In general, for k specifi…

Three friends all in a row

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