Question

# Co-efficient of αt in the expansion of,(α+p)m−1+(α+p)m−2(α+q)+(α+p)m−3(α+q)2+........(α+q)m−1, where α≠−q and p≠q is (α+p)m−1+(α+p)m−2(α+q)+(α+p)m−3(α+q)2+........(α+q)m−1,

A
mCt(ptqt)pq
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B
mCt(pmtqmt)pq
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C
mCt(pt+qt)pq
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D
mCt(pmt+qmt)pq.
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Solution

## The correct option is C mCt(pm−t−qm−t)p−qLet S=(α+p)m−1+(α+p)m−2(α+q)+(α+p)m−3(α+q)2+....+(α+q)m−1As we can see, S is a Geometric Sum with,First term, a=(α+p)m−1,Common Ratio, r=α+qα+p andNumber of terms, n=mSo, Sm=(α+p)m−1(1−(α+qα+p)m)(1−α+qα+p)⇒Sm=(α+q)m−(α+q)mp−qOn doing binomial expansion on Sm,Sm=1p−q(∑mt=0mCtαtpm−t−∑mt=0mCtαtqm−t)From the equation, we can see that,Coefficient of αt=mCt(pm−t−qm−t)p−qSo, Option B is the correct answer.

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