simplest way to write it is have a subquery to determine the count of appointments per client, then group again by that count:

CREATE TABLE #appt (ClientID int, AppointmentDate char(1))

INSERT INTO #appt

VALUES

(1,'x'),

(1,'y'),

(2,'c'),

(3,'y'),

(3,'c'),

(4,'a'),

(4,'b'),

(4,'c')

SELECT ApptCount, COUNT(*) AS ClientCount

FROM (SELECT COUNT(*) AS ApptCount FROM #appt GROUP BY ClientID) ac

GROUP BY ac.ApptCount