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BITSAT2010MathematicsSequences and SeriesActual

The coefficient of (x³² ) in the expansion of ( (x^4- 1 x^3 )¹⁵ ) is:

Options

  1. A( ⁻¹⁵ C ₃ )
  2. B( ¹⁵ C ₄ )
  3. C( ⁻¹⁵ C ₅ )
  4. D( ¹⁵ C ₂ )

Correct answer

B. ( ¹⁵ C ₄ )

Step-by-step solution

We know by Binomial expansion, that ((x+a)^ n ) ( aligned &= ^ n C ₀ x^ n a ^0+ ^ n C ₁ x ^ n -1 a + ^ n C ₂ x ^ n -2 a ^2 &+ ^ n C ₃ x ^ n -3 a ^3+ ^ n C ₄ x ^ n -4 a ^4+ . .+ ^ n C _ n x ^0 a^ n aligned ) Given expansion is ( (x^4- 1 x^3 )¹⁵ ) On comparing we get ( n =15, x = x ^4 ), (a= (- 1 x^3 ) ) ( aligned & (x^4- 1 x^3 )¹⁵= ¹⁵ C ₀ ( x ^4 )¹⁵ (- 1 x ^3 )^0 & + ¹⁵ C ₁ ( x ^4 )¹⁴ (- 1 x ^3 )+ ¹⁵ C ₂ ( x ^4 )¹³ (- 1 x ^3 )^2 & + ¹⁵ C ₃ ( x ^4 )¹² (- 1 x ^3 )^3+ ¹⁵ C ₄ ( x ^4 )¹¹ (- 1 x ^3 )^4+ & T _ r +1 = ¹⁵ C

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