What is the total number of ways of choosing $4$ cards from a pack of $52$ playing cards? In how many of these are the four cards of the same suit?

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 all the letters of the word $'GANGARAM'$ are arranged,then the number of words in which exactly two vowels are together but no two $'G'$ occur together is-

In an examination,there are $5$ multiple choice questions with $3$ choices each,out of which exactly one is correct. There are $3$ marks for each correct answer,$-2$ marks for each wrong answer,and $0$ marks if the question is not attempted. The number of ways a student appearing in the examination can get exactly $5$ marks is:

In how many different ways can three persons $A, B, C$ having $6, 7$ and $8$ one-rupee coins respectively,donate $₹ 10$ collectively?

The number of $6$-letter words,with or without meaning,that can be formed using the letters of the word $\text{MATHS}$ such that any letter that appears in the word must appear at least twice,is $...$