hkcee binomial

hkcee - A. maths - binomial theorem

Q:
Consider P = (1 + x² + x³)ⁿ.
Find the coefficients of x^5 and x^7 in terms of n.



A1:

Rewrite P as (1 + t)^n where t = x^2 + x^3 .
Then P = 1 + n t + C(n,2) t^2 + ..

Note that
t^2 = x^4 + 2x^5 + x^6
t^3 = x^6 + 3x^7 + ..
t^4 = x^8 + ..

Hence,
coeff. of x^5 = 2 C(n,2) = n(n-1)
coeff. of x^7 = 3 C(n,3) = n(n-1)(n-2)/2



A2:

[ 1 + (x^2 +x^3) ]^n
= 1 + n (x^2 + x^3)
　+ C(n,2) (x^4 + 2x^5 + x^6)
　+ C(n,3) (x^6 + 3x^7 + 3x^8 + x^9)
　+ ..

Hence,
coeff. of x^5 = 2 C(n,2) = n(n-1)
coeff. of x^7 = 3 C(n,3) = n(n-1)(n-2)/2