Permutation Qns~ ( try to answer them with logical workings)?

The positions of nine trees which are to be planted along the sides of a road, four on the north side and five on the south side, are shown in the figure.





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a) Find the number of ways in which this can done if the trees are all of different species.





b) If the trees in (a) are planted at random, find the probability that two particular trees are next to each other on the same side of the road.





c) If there are three angsana, four cocoa and two palm, find the number of different ways in which these could be planted assuming that trees of the same species are identical.





d) If there are twenty different spices of trees (including angsana, cocoa and palm) to be selected, find the number of ways in which this can be done if angsana, cocoa and palm are to be included in any of the nine positions.

Permutation Qns~ ( try to answer them with logical workings)?
a) Each is different, so there a 9! ways to order them (pick a tree for first spot, 9 ways, pick tree for second spot, 8 ways, etc)





d) This is a variation of a circular permutation. You have to first find out how many ways there are to arrange the two trees so they are next to each other (2*8) and the arrange the rest (7!). So all together there are 2*8! ways, so it's 2*8!/9!, which is 2/9





c) it is 9!/(3!*4!*2!). Easiest way to visualize this is you arrange the 9, and then 'get rid' of all the identical countings by dividing the amount of ways you can arrange them.





d) Well, Now you are selecting amoungst 20 species. You know 3 of them are Angsana, cocoa, and palm, you can choose those positions first (9*8*7 ways), and then choose 6 more speciest ( 17 choose 6 ways ) and then arrange them (6! ways), so putting it all together, the total is 9*8*7*( 17 C 6) *6! = 9! * (17 C 6)