Finding number of child processes of a particular process

ps -eo ppid= | grep -Fwc $pid

If your grep does not support -w:

ps -eo ppid= | tr -d '[:blank:]' | grep -Fxc $pid

or

ps -eo ppid= | awk '$1==ppid {++i} END {print i+0}' ppid=$pid

or (clobbering the positional parameters)

set $(ps -eo ppid=); echo $#

Note that this is not atomic, so the count may be wrong if some processes die and others get spawned in the short span of time it takes to gather the data.

Try ps --ppid ${pid} --no-headers | wc --lines.