There are $3$ applicants for a scholarship in Physics,$5$ for Mathematics,and $4$ for Chemistry. In how many different ways can one of these scholarships be awarded?

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 $.......$

The number of all $3$-digit numbers $abc$ (in base $10$) for which $(a \times b \times c) + (a \times b) + (b \times c) + (c \times a) + a + b + c = 29$ is:

$A$ question paper has $3$ parts and each part contains $4$ questions. The number of different ways in which a candidate can answer $8$ questions choosing at least two from each part is

The number of positive integers less than $10000$ which contain the digit $5$ at least once is

The number of ways in which an arrangement of $4$ letters of the word $PROPORTION$ can be made is