Midrasch zu Wajikra 16:24

3) R. Yehudah said: Whence are derived the five immersions and the ten lustrations (washing of hands and feet) that the high-priest performs on that day? From (Vayikra 16:23) "Then Aaron shall come to the tent of meeting, and he shall take off the linen garments which he put on when he went into the holy place … (Vayikra 16:24) And he shall bathe his flesh in water in a holy place, and he shall put on his (eight everyday) garments." From here we derive that all transitions from one service to another requires immersion — whence we derive that he performed five immersions and ten lustrations on that day.

8) "And he shall bathe his flesh in water in a holy place": In the Beth Haparvah quarter (in the south of the azarah). And five immersions took place there on that day, all in the holy (area) of the Beth Haparvah — aside from the first that he performed (for the slaughtering of the tamid) every day at the Water Gate. (Vayikra 16:24) "and he shall go out and offer up his burnt-offering and the burnt-offering of the people, and he shall make atonement for himself and for the people.": His burnt-offering precedes that of the people, and his atonement precedes that of the people.