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With grep:

cat poem.txt \
  | grep -Evi -e '\<an\>.*\<the\>' -e '\<the\>.*\<an\>' \
  | grep -Eci -e '\<(an|the)\>'

This counts the matched lines. You can find an alternative syntax which counts the total number of matches down below.

Breakdown:

The frist grep command filters out all lines containing both 'an' and 'the'. The second grep command counts those lines, containing either 'an' or 'the'.

If you remove the c from the second grep's -Eci, you will see all matches highlighted.

Details:

• The -E option enables extended expression syntax (ERE) for grep.

• The -i option tells grep to match case-insensitive

• The -v option tells grep to invert the result (i.e. match lines not containing the pattern)

• The -c option tells grep to output the number of matched lines instead of the lines themselves

• The patterns:

  1. \< matches the beginning of a word (thanks @glenn-jackman)
  2. \> matches the end of a word (thanks @glenn-jackman)

  --> That way we can make sure to not match words containing 'the' or 'an' (like 'pan')

  3. grep -Evi -e '\<an\>.*\<the\>' thus matches all lines not containing 'an ... the' ~(Note: I did purposefully not include the case "an the" ('the' directly following on 'an' because it is an unlikely case and I wanted to keep the pattern simple. It could, of course, easily be added)~.
  4. Similarly, grep -Evi -e '\<the\>.*\<an\>' matches all lines not containing 'the ... an'
  5. grep -Evi -e '\<an\>.*\<the\>' -e '\<the.*an\>' is the combination of the 3. and 4.
  6. grep -Eci -e '\<(an|the)\>' matches all lines containing either 'an' or 'the' (surrounded by whitespace or start/end of line) and prints the number of matched lines

EDIT 1: Use \< and \> instead of ( |^) and ( |$), as suggested by @glenn-jackman

EDIT 2: In order to count the number of matches instead of the number of matched lines, use the following expression:

cat poem.txt \
  | grep -Evi -e '\<an\>.*\<the\>' -e '\<the\>.*\<an\>' \
  | grep -Eio -e '\<(an|the)\>' \
  | wc -l

This uses the -o option of grep, which prints every match in a separate line (and nothing else) and then wc -l to count the lines.