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(1) Out of $8$ sailors on a boat , $3$ can work only on one side and $2$ only on other side.

Then the number of ways the sailors can be arranged on a boat , is

(2) Passengers are to travel by a double decked bus which can accommodate $13$ in the upper deck

and $7$ in the lower deck. the no. of ways that they can be distributed if $5$ refuse to sit in the upper

deck and $8$ refuse to sit in the lower deck is.

$\bf{My\; Try::}$ For $(1)$...First we will select $3$ out of $8$ sailors, This can be done in $\displaystyle = \binom{8}{3}$ ways.

Now we will arrange $2$ sailors out of $5$ sailors, This can be done in $\displaystyle = \binom{5}{2}$

Now total no. of ways $\displaystyle = \binom{8}{3}\cdot \binom{5}{2}\cdot 3!\cdot 2!$

But my answer is wrong,

Now How can i solve it

please help me

Thanks

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  • $\begingroup$ For the sailor problem, would need some information: (i) do we want $4$ on each side? If it's a rowboat we probably do. (ii) does the order of the sailors on the two sides matter? If the answer is yes to both questions, then the ultimate answer is $\binom{3}{1}(4!)^2$. $\endgroup$ – André Nicolas May 22 '14 at 6:29
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You're interpreting the question wrong. It means that 3 particular people must be on the left and 2 other particular people must be on the right. You're not free to choose those 5 people. Instead you have to arrange all 8 in some way to satisfy those 5 people. But then you also need to see what the problem says about how many must be on each side. The second problem can be solved in exactly the same way.

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(1) Out of 8 sailors on a boat , 3 can work only on one side and 2 only on other side.

Then the number of ways the sailors can be arranged on a boat , is

You have two sides to send the sailors. Send the 3 to side A, send the other 2 to side B. They cannot be moved. The remaining 3 sailors can each be assigned to either side. How many ways can you rearrange these sailors?

It depends.

We do not know anything about the boat other than it has two sides for sailors to work. Is there a limit to how many sailors work a side? (If so, what?) Can the sailors move about on each side, or do they have fixed stations?

If there is no limit, and no fixed stations, then there are $8$ to select sides for the three remaining sailors. ($2^3$) Choose a side for each sailor.

If there is a limit of 4 sailors to a side, but no fixed stations, then there are but $3$ ways to arrange the remaining three sailors. (${3\choose 1}$) Choose one sailor to join side A, send the others to side B.

If there is a limit of 4 fixed stations on each side, then there are $1728$ ways to arrange the remaining three sailors. (${3\choose 1}\times(4!)^2$) Choose which sailor adds to side A, then assign stations to each sailor on each side.

What if there are three fixed station on each side, and two in the middle (or one each aft and bow)? ...


(2) Passengers are to travel by a double decked bus which can accommodate 13 in the upper deck and 7 in the lower deck. the no. of ways that they can be distributed if 5 refuse to sit in the upper deck and 8 refuse to sit in the lower deck is?

5 and 8 of how many passengers in total? (Ie: Do we have a full bus load or will there be empty seats?) Also, again, do you care about exact seating, or just decks?

Answer these issues, then solve as above.

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