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In my text book the question is as follows:

Find the way in which $5$ persons can sit in a row if two insist on sitting next to each other.

They give the answer as $48$. I fail to understand how they got there because when I try the standard formula of $P = \frac{n!}{(n-r)!}$ where $n = 5$ and $r = 4$ I get $120$. If I try the formula that considers repetition $P = n!/n1! * n2!...nx!$, I get $60$, counting the two that insist on sitting together as a repetition.

Please explain what it is I don't understand.

Kamil Jarosz
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user337046
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1 Answers1

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Let the people be $p_1,p_2,\ldots p_5.$ Let's say $p_1$ and $p_2$ wish to sit together,

So we club $p_1, p_2$ and treat them like a single person,

So now we have $4$ people, the number of ways to seat them will be $4!$, but also notice that in all arrangements we have to consider the permutation of $p_1,p_2$ and $p_2,p_1$ as distinct, thereby giving $4! \times 2! =48 $ ways

Nikunj
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  • I see... So this is not a permutation of n objects taken r at a time, but a permutation of n objects taken ALL at a time? But with consideration of p1 p2 and p2 p1? as sort of a collection within a collection? – user337046 May 04 '16 at 19:33
  • @user337046 Try to do this manually with lesser people, say $3$ and you'll see why the arrangement $p_1,p_2$ and $p_2,p_1$ matters! – Nikunj May 04 '16 at 20:14