I interpreted this question a little differently
"Each sequence cannot begin with L and cannot end with Q "
P(6,4) = 360 possibile sequences
Subtract out the ones that begin with an L and end with a Q
L _ _ Q
The other 2 letters can be chosen in C(4,2) ways = 6 ways and each of these can be arranged in 2 ways =
6 * 2 = 12 words beginning with L and ending with Q
So
360 - 12 =
348 sequences