Find the number of ways in which 4 letters can be selected from the 9 letters of the word ALGORITHM, given at least one vowel is included. My solution:
3 cases:
1 vowel:
$6P3 \times 3C1 = 360 $
2 vowels:
$6P2 \times 3C2 \times 2! = 180 $
3 vowels:
$6P1 \times 3! = 36 $
Total number of ways: 360 + 180 + 36 = 576
Am I correct?

Using the Inclusion-Exclusion principle, we count the number of combinations that use $1$ vowel, subtract those that use $2$, and add those that use all $3$.
The generating function for this is
$$[x^4]:(1-(1-x)^3)\frac{1}{(1-x)^6}$$
where
$$1-(1-x)^3=\binom{3}{1}x-\binom{3}{1}x^2+x^3$$
$$\frac{1}{(1-x)^6}=\binom{5}{5}+\binom{6}{5}x+\binom{7}{5}x^2+\binom{8}{5}x^3+\dots$$
and gives $3\cdot56-3\cdot21+6=111$.
Also, the GF can be re-arranged to give
$$[x^4]:\frac{1}{(1-x)^6}-\frac{(1-x)^3}{(1-x)^6}$$
$$[x^4]:\frac{1}{(1-x)^6}-\frac{1}{(1-x)^3}$$
$$=\binom{9}{5}-\binom{6}{2}$$
$$=111$$
