In 12 hrs. the hands of the clock will be perpendicular to each other for 22 times as in 1 hr. time the hands of the clock wil be at rt. angles for 2 times but between 1 1'o clock and 1'o clock they will be at rt. angles only twice.
Therefore, in 24 hrs. they will be at right angles 44 times.
Edit: Thank you Robin for providing an alternative solution in the comments.