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Question

In the expansion of (713+1119)6561
  1. There are exactly 730 rational terms
  2. There are exactly 5831 irrational terms 
  3. The term which involves greatest binomial coefficients is irrational 
  4. The term which involves greatest binomial coefficients is rational 


Solution

The correct options are
A There are exactly 730 rational terms
C The term which involves greatest binomial coefficients is irrational 
General term is  6561Cr(76561r3)11r9.
Terms will be rational if r is a multiple of 9
 r=0,9,18,27,....,6561.
Now 6561=0+(n1)9n=730
Hence, there are 730 rational terms. The greatest binomial coefficients are  6561C656112 and 6561C6561+12
6561C3280 and 6561C3281
Now, 3280 are 3281 are not a multiple of 3; hence, both terms involving greatest binomial coefficients are irrational. 

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