How many permutations can be made by using all the letters of the word $MATHEMAGICA$?

The number of ways in which $3$ children can distribute $10$ tickets out of $15$ consecutively numbered tickets among themselves such that they get consecutive blocks of $5, 3$ and $2$ tickets is:

In a group of $6$ boys and $4$ girls,four children are to be selected. In how many different ways can they be selected such that at least one boy should be there?

How many different ways can the letters in the word $ATTEND$ be arranged?

The number of different words that can be formed out of the letters of the word $MORADABAD$ taken four at a time is

$^nC_r + ^{n-1}C_r + ...... + ^rC_r$ =