Respuesta :
Using polynomial expansion, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power is 14
According to the question,
For some particular value of N, when [tex](a + b +c + d+ 1)^N[/tex] is expanded and like terms are combined, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power.
Polynomial expansion (a + b)ⁿ = Cₙ⁰ aⁿ + Cₙⁿ⁻¹ aⁿ⁻¹ b + ....+ Cₙⁿbⁿ
(a + b + c + d + 1)ⁿ = ...Ka[tex]a^{X_{a} } b^{X_{b} } c^{X_{c} }d^{X_{d} }1^{X_{1} }[/tex]...
[tex]X_{a}+X_{b} +X_{c} + X_{d} +X_{1} = N[/tex]
[tex]X_{a} > 1 X_{b} > 1 X_{c} > 1 X_{d} > 1 X_{1} > 0[/tex]
Positive integer solution
[tex]X_{a}+X_{b} +X_{c} + X_{d} +X_{1} = N[/tex]+1
Non negative integer solution
[tex]X_{a}+X_{b} +X_{c} + X_{d} +X_{1} = N[/tex] - 4
[tex]C_{N}^4[/tex] = 1001
1001 = 7.11.13 after some trial and error we find [tex]C_{14}^4 = 1001[/tex] that is N is equal to 14.
hence, using polynomial expansion, the resulting expression contains exactly 1001 terms that include all four variables a, b, c, and d, each to some positive power is 14
Learn more about polynomial expansion here
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