I _ _ _ L

We can fill in the blanks with 3 letters taken from the set {D,R,I,L,N,G}

Order matters, so that's the permutations of 6 things taken 3 at a time.

P(6,3) or some people write 6P3.  That's 6×5×4 = 120 ways.

Here they all are:

IDRIL IDRLL IDRNL IDRGL IDIRL IDILL IDINL IDIGL IDLRL IDLIL IDLIL
IDLNL IDLGL IDNRL IDNIL IDNLL IDNGL IDGRL IDGIL IDGLL IDGNL IDGNL
IRDIL IRDLL IRDNL IRDGL IRIDL IRILL IRINL IRIGL IRLDL IRLIL IRLIL
IRLNL IRLGL IRNDL IRNIL IRNLL IRNGL IRGDL IRGIL IRGLL IRGNL IRGNL
IIDRL IIDLL IIDNL IIDGL IIRDL IIRLL IIRNL IIRGL IILDL IILRL IILRL
IILNL IILGL IINDL IINRL IINLL IINGL IIGDL IIGRL IIGLL IIGNL IIGNL
ILDRL ILDIL ILDNL ILDGL ILRDL ILRIL ILRNL ILRGL ILIDL ILIRL ILIRL
ILINL ILIGL ILNDL ILNRL ILNIL ILNGL ILGDL ILGRL ILGIL ILGNL ILGNL
INDRL INDIL INDLL INDGL INRDL INRIL INRLL INRGL INIDL INIRL INIRL
INILL INIGL INLDL INLRL INLIL INLGL INGDL INGRL INGIL INGLL INGLL
IGDRL IGDIL IGDLL IGDNL IGRDL IGRIL IGRLL IGRNL IGIDL IGIRL IGIRL
IGILL IGINL IGLDL IGLRL IGLIL IGLNL IGNDL IGNRL IGNIL IGNLL IGNLL

I _ _ _ L

Then we can fill in the blanks with 3 letters taken from the set {D,R,N,G}

Order matters, so that's the permutations of 4 things taken 3 at a time.

P(4,3) or some people write 4P3.  That's 4×3×2 = 24 ways.

Here they are:

IDRNL IDRGL IDNRL IDNGL IDGRL IDGNL IRDNL IRDGL IRNDL IRNGL IRNGL
IRGDL IRGNL INDRL INDGL INRDL INRGL INGDL INGRL IGDRL IGDNL IGDNL
IGRDL IGRNL IGNDL IGNRL

I _ _ _ L

We must use an I and an L since they each occur twice in DRILLING.

That's the permutations of 3 blanks choose 2 to put the I and L.
P(3,2) = 3×2 = 6 ways.

Then we can choose the other letter from the set {D,R,N,G).
That's 4 ways.

So we multiply 6×4 = 24 ways.

Here they are:

ILIDL ILIRL ILINL ILIGL 
ILDIL ILRIL ILNIL ILGIL 
IILDL IILRL IILNL IILGL 
IIDLL IIRLL IINLL IIGLL   
IDLIL IRLIL INLIL IGLIL 
IDILL IRILL INILL IGILL 

Edwin