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Question

In the expansion of x3-1x2n,nNif the sum of the coefficients of x5 and x10 is 0, then n is equal to


A

25

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B

20

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C

15

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D

None of these

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Solution

The correct option is C

15


Determine the value of n

Step 1: Determine the coefficients

We know that in the expansion of (a+b)n, rth term is given as : Tr+1=nCran-rbr

Tr+1=ncr(x3)nr1x2rTr+1=ncrx3n3r(1)rx2rTr+1=ncrx3n5r(1)r...(i)

Now for coefficients of x5 substituting 3n-5r=5 we get,

r=3n-55....(ii)

The coefficients of x5 is

nC3n55(1)3n55....(iii)[From(i)&(ii)]

Similarly coefficients of x10 substituting 3n-5r=10we get

r=3n-105....(iv)

The coefficients of x10 is

C3n105(1)3n105....(v)[Ftom(i)&(iv)]

Step 2: Determine the value of n

According to question: coefficients of x5 + coefficients x10 =0

nC3n55(1)3n55+nC3n105(1)3n105=0nC3n55nC3n105(1)3n105=03n55+3n105=n6n155=n6n15=5nn=15

Hence, option C is correct .


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