Count the permutations ili2i3i4i5i6 of {1,2,3,4,5,6}, where i11, i21,2, i33,4, i43.

A chess master has 30 days to prepare for a tournament. She will play no more than 45 games. She also wishes to play at least 1 games per day. Show that no matter how she schedules her preparation, there is a succession of days during which she will have played exactly 14 games.