A ratio of the 5th term from the beginning to the 5th term from the end in the binomial expansion of 213+12(3)1310 is

Since, rth term from the end in the expansion of a binomial (r +α)n is same as the (n - r +2)th term from the beginning in the expansion of same binomial.∴ Required ratio =T5T10−5+2=T5T7=T4+1T6+1⇒ T5T10−5+2= 10C421/310−412(3)1/34 10C621/310−612(3)1/36∵Tr+1=nCrxn−rar∵10C4=10C6=26/32(3)1/3624/32(3)1/34=26/3−4/32(3)1/36−4=223⋅22⋅323=4(6)23=4(36)1/3So, the required ratio is. 4(36)1/3:1