(a)[1]
Find how many distinct ways the $9$ letters in HAPPINESS can be arranged.
(b)[3]
Find the number of distinct arrangements of the $9$ letters in HAPPINESS for which the first and last letters are different.
(c)[4]
Find the number of distinct arrangements of the 9 letters in HAPPINESS in which the two Ps are adjacent and there are exactly two letters between the two Ss.
(d)[3]
At random, the 9 letters in HAPPINESS are split into a group of 5 and a group of 4. Find the probability that both Ps end up in one group and both Ss in the other group.