How many numbers of $6$ digits can be formed from the digits of the number $112233$

$^{14}{C_4} + \sum\limits_{j = 1}^4 {^{18 - j}{C_3}} $ is equal to

Find the number of words with or without meaning which can be made using all the letters of the word $AGAIN$. If these words are written as in a dictionary, what will be the $50^{\text {th }}$ word?

A boy needs to select five courses from $12$ available courses, out of which $5$ courses are language courses. If he can choose at most two language courses, then the number of ways he can choose five courses is

The number of ways in which four letters of the word $‘MATHEMATICS$’ can be arranged is given by

A scientific committee is to be formed from $6$ Indians and $8$ foreigners, which includes at least $2$ Indians and double the number of foreigners as Indians. Then the number of ways, the committee can be formed, is