The number of four-digit numbers that can be formed using the digits $1, 2, 3, 4,$ and $5$ such that at least two digits are identical is

The number of four-digit numbers formed by using the digits $0, 2, 4, 5$ (without repetition) which are not divisible by $5$ is:

The value of the expression ${ }^{47} C_4 + \sum_{j=1}^5 { }^{52-j} C_3$ is

If the letters of the word $ATTEMPT$ are rearranged randomly,what is the probability that all $T$'s occur together?

All the arrangements,with or without meaning,of the word $FARMER$ are written excluding any word that has two $R$ appearing together. The arrangements are listed serially in the alphabetic order as in the English dictionary. Then the serial number of the word $FARMER$ in this list is .... .

How many words can be formed by taking two identical and two distinct letters from the letters of the word $COMBINATION$?