M2的數學問題 (20點)

(1) 1/(n+2)! - 1/(n+1)! - 1/n!
= (1/n!){1/[(n+1)(n+2)] - 1/(n+1) - 1}
= (1/n!)[1 - (n+2) - (n+1)(n+2)]/(n+1)(n+2)
= [1/(n+2)!](1 - n - 2 - n^2 - 3n - 2)
= [1/(n+2)!](-3 - 4n - n^2)
= -(n+1)(n+3) / (n+2)!
(2) (r+s+t)Cr * (s+t)Cs
= (r+s+t)!/[r!(s+t)!] * (s+t)!/[s!t!]
= (r+s+t)!/[r!s!t!]
(3) (n+1)C3 - (n-1)C3
= (n+1)!/[3!(n-2)!] - (n-1)!/[3!(n-4)!]
= (n-1)!/[3!(n-4)!]{n(n+1)/(n-2)(n-3) - 1}
= (n-1)!/[3!(n-4)!]{n(n+1) - (n-2)(n-3)}/(n-2)(n-3)
= (n-1)!/[3!(n-2)!][n^2 + n - n^2 +5n - 6]
= (n-1)!/[3!(n-2)!][6n - 6]
= (n-1)(n-1)!/(n-2)!
= (n-1)^2