SOLUTION: How many 7-letter arrangements can be created using all of the letters in the word FLOWERS, if the first letter must be an "R" and the last letter must be a consonant?

Choose FIRST the letter posititions which have the most 
restrictive attributes.

The first letter position has the most restrictive attribute.
It must be R. We can choose it only 1 way, as R.

That leaves letters F,L,O,W,E,S to place in letter positions
 
The remaining letter position with the most restrictive attribute is
the last, 7th, letter, which must be a consonant.  Only consonants 
F,L,W,S are available.  That's 4 ways.

Regardless of which consonant we chose for the 7th letter position, there
are 5 letters left, and none of the other positions are restrictive,
so we might as well take them left to right:

We choose the 2nd letter position to have any one of the 5 remaining 
letters.

That leaves 4 letters. 

We choose the 3rd letter position to have any one of the 4 remaining 
letters.  

That leaves 3 letters. 

We choose the 4th letter position to have any one of the 3 remaining 
letters.

That leaves 2 letters. 

We choose the 5th letter position to have either one of the 2 remaining 
letters.

That leaves only 1 letter. 

We choose the 6th letter position to have that only one remaining 
letter.

Summary:

1st letter position can be chosen 1 way.
7th letter position can be chosen any one of 4 ways.
2nd letter position can be chosen any one of 5 ways.
3rd letter position can be chosen any one of 4 ways.
4th letter position can be chosen any one of 3 ways.
5th letter position can be chosen either of 2 ways.
6th letter position can be chosen only 1 way.

Answer: 1×4×5×4×3×2×1 = 480 ways

Edwin