Paragraph: In the expansion of (x^3- 1 x^2 )^n , where n is a positive integer, the sum of the coefficients of x^5 and x¹⁰ is 0. Question: What is n equal to?

Step-by-step solution

aligned & (x^3- 1 x^2 )^n & = ^n C ₀ (x^3 )^n- ^n C ₁ (x^3 )^ n-1 ( 1 x^2 )+ ^n C ₂ (x^3 )^ n-2 ( 1 x^2 )^2 & - ^n C ₃ (x^3 )^ n-3 ( 1 x^2 )^3+ ^n C ₄ (x^3 )^ n-4 ( 1 x^2 )^4- ^n C ₅ (x^3 )^ n-5 & ( 1 x^2 )^5+ ^n C ₆ (x^3 )^ n-6 ( 1 x^2 )^6- ^n C ₇ (x^3 )^ n-7 ( 1 x^2 )^7 & + ^n C ₈ (x^3 )^ n-8 ( 1 x^2 )^8- ^n C ₉ (x^3 )^ n-9 ( 1 x^2 )^9 aligned aligned & ^n C ₈= ^n C ₇ n=8+7 n=15 aligned Hence, n=15 .