permutation and combination

Question

in how many ways 10 identical blue marbles and 5 identical green marbles be arranged in a row so that no two green marbles are together??

Answer 1

First we place the 10 blue balls in 1 way because they are all identical :

 __B1__  B2__B3__B4__B5__B6__B7__B8__B9__B10__

Since no 2 green balls   are together only place we have is to place the green balls in between blue balls . There are 11 available places to place the green balls : Therefore we can select any 5 places as 11C5 . Then we arrange the 5 balls in 1 way because they are identical .

Therefore the answer is 11C5 = 462.

Comments

what if they are not identical . then what will be the case

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If they are not identical you do multiply 11C5 with 10! and 5! becoz blue balls can arrange among themselves in 10! ways  and 5 green balls in 5! different ways .

Therefore : 11C5*10!*5! would have been the answer .

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We have already place 5 green balls in 11C5 ways, then why we again multiplying 5! ?

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Why isnt it 11P5 coz we are arranging the 5 green balls in restricted places of 11 poistions.And the blue balls are suposed to arrange in 10! ways.now since they are identical it is 10!/10!.Plz clarify someone,.