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Question

The sum mi=0(10i)(20mi), where (pq)=0 if p>q, is maximum when m is equal to


  1. 5

  2. 10

  3. 15

  4. 20


Solution

The correct option is C

15


mi=0(10i)(20mi) is the coefficient of xm in the expansion of (1+x)10(x+1)20,
  mi=0(10i)(20mi) is the coefficient of xm in the expansion of (1+x)30
i.e.   mi=0(10i)(20mi)=30Cm=(30m)         . . . (i)
and we know that, (nr) is maximum, when
(nr)max={r=n2if n ϵ evenr=n±12if n ϵ odd
Hence, (30m) is maximum when m = 15

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