Interesting Pat hand puzzle

[TyFuji]

Simple logic problem: Is there a combination of 25 standard playing cards for which there is no possible way to make 5 separate pat hands, using all 25 cards? For this problem, the only pat hands are straights, flushes, and full houses.

My first thought is that there are no straights which do not contain T's or 5's. Any thoughts?

[TomCowley]

<font color="white">
2s 2h 2d
3s 3h 3d
4s 4h 4d
5s 5h 5d
7s 7h 7d
8s 8h 8d
9s 9h 9d
Ts Th Td
Qc

25 cards, and you can't make any pat hand with Qc in it. QED. </font>