How many words of $7$ letters can be formed by selecting $4$ consonants and $3$ vowels from $6$ consonants and $5$ vowels?

How many numbers between $5000$ and $10000$ can be formed using the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ if no digit is repeated?

If $3$ books on Physics,$2$ books on Chemistry,and $4$ books on Mathematics are to be arranged on a shelf so that all the Physics books are together and all the Mathematics books are together,then the number of such arrangements is

Let $b_{1} b_{2} b_{3} b_{4}$ be a $4$-element permutation with $b_{i} \in \{1, 2, 3, \ldots, 100\}$ for $1 \leq i \leq 4$ and $b_{i} \neq b_{j}$ for $i \neq j$,such that either $b_{1}, b_{2}, b_{3}$ are consecutive integers or $b_{2}, b_{3}, b_{4}$ are consecutive integers. Find the number of such permutations.

$10$ different letters of the English alphabet are given. Out of these letters,words of $5$ letters are formed. How many words can be formed if at least one letter is repeated?

If $^{12}P_r = 1320$,then $r$ is equal to