Out of $7$ consonants and $4$ vowels, words are formed each having $3$ consonants and $2$ vowels. The number of such words that can be formed is

Let $S$ be the set of all passwords which are $6$ to $8$ characters long,where each character is either an alphabet from $\{A, B, C, D, E\}$ or a number from $\{1, 2, 3, 4, 5\}$ with the repetition of characters allowed. If the number of passwords in $S$ whose at least one character is a number from $\{1, 2, 3, 4, 5\}$ is $\alpha \times 5^{6}$,then $\alpha$ is equal to $.......$

If ${}^n P_r = 30240$ and ${}^n C_r = 252$,then the ordered pair $(n, r)$ is equal to

$^{29}C_5 + \sum_{r=0}^{4} {^{(33-r)}C_4} =$

If all the letters of the word $ACADEMICIAN$ are permuted in all possible ways,then the number of permutations in which no two $A$'s are together and all the consonants are together is:

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