A student was asked to make all the possible combinations of the word POTATO, where each word should begin and end with O. How many words can he form?

The "O's" at the beginning and end are indistinguishable

So.....we have 4!  = 24 possible arrangements  of the middle letters "PTAT'

But the "T's"  are also indistinguishable....so we need to divide by  2  = 12 distinguishable words....