Answer :
Tom and John are to sit in a row of 6 seats with 4 other people. Number of ways for 6 people to sit if Tom sits to the left of John (but not necessarily directly beside him) is given by N = John on extreme right seat (#6) P(5, 1)*P(4, 4) + John on seat #5 P(4, 1)*P(4, 4) + John on seat #4 P(3, 1)*P(4, 4) + John on seat #3 P(2, 1)*P(4, 4) + John on seat #2 P(1, 1)*P(4, 4) = P(4, 4)[5 + 4 + 3 + 2 + 1] = 24*15 = 360 ANSWER P(4, 4) is the number of ways seating 4 others on remaining 4 seats when John has seated and Tom has seated on his left in P(5, 1), P(4, 1), P(3, 1), P(2, 1), P(1,1) ways.
You can use the fact that the only thing that matters is sitting position of tom and john.
There are total 55 ways for the 11 people to sit if tom sits to the left of john
How to find the number of ways all these 11 people can sit tom sits to the left of john?
Let we enumerate the seats from 1 to 11.
Let john sits on [tex]k^{th}[/tex] seat.
Then there are 11-k seats available on the left of john for tom to sit. And tom can sit on those 11-k seats in [tex]^{11-k}C_1 = 11 - k[/tex] ways.
Since k will take values from 1 to 11, we have:
[tex]\text{Total ways} = \sum_{k=1}^{11} (11-k) = 11 \times 11 - \sum_{k=1}^{11}k = 121 - \dfrac{11(11+1)}{2} = 121 - 66\\\text{Total ways} = 55[/tex]
(since adding 1 to n integers is done by [tex]\dfrac{n(n+1)}{2}[/tex] formula, and we used it for n = 11)
Thus,
There are total 55 ways for the 11 people to sit if tom sits to the left of john
Learn more about combinations here:
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