How many six-letter sequences are possible that use the letters c, i, c, c, i, y?

Answer 1

notice we got 3 c's and 2 i's

suppose the c's and i's are all different, therefore distinguishable.
(pretend they are all of a different colour)

then they can be arranged in 6! ways.

but we CAN'T tell the c's apart, so we have counted 3! of them that look alike.
similarly we counted 2! of them with i's that look alike,
these have to be divided out.
so the number of ways is 6!/(3!2!) = 60

this is the general procedure for these types of questions.
first take the factorial of the total, then divide by the factorials of each duplication count
e.g. aaabbcccc
can be arranged in 9!/(3!2!4!)ways.