Question

Number of combination of different things taking r things at a time when,

• A. $n-P-QCr-P$
• B. $n-PCr-P$
• C. $n-PCr$

Answer 1

1. $P$ things should always be included implies that no. of things left to be selected $=n-P$
and from $n-P$ things we have to select $r$ things when $P$ things are already selected.
this implies that things left to be selected are $r-P$
therefore, answer is
= $n-PCr-P$
2. when $P$ things are always excluded then no. of things left from where we can select $r$ things are $n-P$ things
therefore, answer is
$n-PCr$
3. similar to our 1st case and 2nd case. $P$ is always included and $Q$ is always excluded implies that out of the $n-P-Q$ things left to be selected, we have to select $r-P$ things since $P$ things are already selected.
therefore, correct answer is
= $n-P-QCr-P$