f.4 biomial expansion20marks!!

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