It wouldn't save characters, but you can eliminate the initial cut if you sort and uniq -c by field:
sort -t $'\t' -k2 file.txt | uniq -f2 -s1 -c
That at least removes a command from the chain. You could also combine the last head and cut with a simple awk one-liner:
awk '{if(NR<11)print $3}'
This is both longer and less simple, but again saves a command.