# Permutation and combination

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The number of ways can 10 balls be selected from urn contain 10 identical red balls 5 identical green balls and 3 identical blue balls ?

retagged
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its  24, 12c8-6c4-8c6+2c0

calculation mistakes :P
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I am getting 16 what is the answer?
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It will be obviously greater than 3 @vyas
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24 is the given answer. They have used generating functions but i m not able to understand anything from that.

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@minal mam in place of 12C8 , 12C10 will be ome
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x+y+z =10

Number of way will be 10+3-1 C 3-1 = 66

Of all these 66 cases some are not valid

The conditions are x>=11, y>=6 , z>=4

Now x >=11 will be for 0 cases

Now y>=6 will be for x+y'+z = 10-6=4 so 4+3-1C 3-1= 15

Now z>=4 will be for x+y+z'= 10-4=6 so 6+3-1C3-1 = 28

But the conditions of y and z can be not valid when y=6 and z=4 is 1 way , this should be added as it will be subtracted for both y and z twice so the answer is 66-15-28+1= 24
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as i know whenever more than 1 upper constraints given then we  use generating function .. complement method is not correct for more than 1 upper constraints
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Why (n+r-1)C(r-1) ??? When to use this formula??
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@vyas why complement will not be true,..can u say by giving example..I think this is correct

@jain that is for number of non negative integer solutions for the equation
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when upper constraints are not given .. or only one upper constraints given then with adjustment (complement method ) we use .. in this case it is giving correct ans becoz x^11 gives 0 ,  but more than 1 upper constraints are given then use generating fun .
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That is why I added the common part which got subtracted twice
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x+y+z=15 , where 0<=x,y,z<=10

try this
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106?
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from generating 91 correct one  , and without it 620 i guess which is wrong ...
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@sonam vyas Give the recurrence relation solution if you have it. It would be helpful.
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## Related questions

1
474 views
9 different books are to be arranged on a bookshelf. 4 of these books were written by Shakespeare, 2 by Dickens, and 3 by Conrad. How many possible permutations are there if the books by Conrad must be separated from one another?