Number the letters

A1 R1 C1 H1
R2 R3 C2 H2
C3 C4 C5 H3
H4 H5 H6 H7

Let's spell ARCH backward as HCRA. From each R there will be only 1 choice
for A which is the only A, namely A1.  So we only need to spell HCR. 

From H1 there are 2 choices for C, C1 and C2. and for each C there are 2 choices
for R, R1 and R3.  That's (2)(2)=4 ways.  By symmetry there's also 4 ways for 
H4.  That's 8 ways.

From H2 there are 3 choices for C, C1, C2, C5.
For 2 of those choices, C1 and C2, there are 2 choices for R, R1, R3. That's 
(2)(2)=4. For C5 there is only 1 choice for R, R3.  That's 5. By symmetry there
are also 5 ways for H5.  That's 10 ways.

From H3 there are 2 choices for C, C2 and C5.
For 1 of those choices, C2, there are 2 choices for R, R1 and R3.
For C5 there is only 1 choice for R, R3.  That's 3. By symmetry there are also 3
ways for H6.  That's 6 ways.

From H7, there is only 1 way to spell HCR, H7-C5-R3 

Answer: 8+10+6+1 = 25

Edwin