doubt

Question

Given 3 characters a, b, c. Find the number of strings of length n that can be formed from these 3 characters. Given that : we can use ‘a’ as many times as we want, ‘b’ maximum once, and ‘c’ maximum twice.

Comments

$\frac{n^{3}+3n+2}{2}$ ??

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I think answer should be

case 1: no b's and c's then its 1

case 2: only one b's and rest all a's then its n!/(n-1)!

case 3: only one c's and rest all a's then its n!/(n-1)!

case 4: only one b's and c's and rest all a's then its n!/(n-2)!

case 5: only two c's and rest all a's then its n!/((n-2)!*2!)

case 6: only two c's and one b's and rest all are a's then its n!/((n-3)!*2!)

so total number of string will be sum of all the above cases.

i.e.  1+n!/(n-1)! + n!/(n-1)! + n!/(n-2)! + n!/((n-2)!*2!) + n!/((n-3)!*2!)