Fundamental Principle of Counting

वह कौनसी संख्या है जिसको अपने में ही $20$बार जोड़ने से परिणाम $861$आता है ?

छः लगातार आने वाली प्राकृत संख्याओं में से यदि तीन का योगफल $27$है तो दूसरी तीन का योगफल क्या होगा ?

The number of six digit number formed by using the digits $\left\{1,2,3,4,5,6\right\}$ which are divisible by $6$ (repetition is not allowed)

Using the number $1,2,3,....,7$, total numbers of $7$ digit number which does not contain string $154$ or $2367$ is, (repetition is not allowed)

If $n$ is a factor of $72$ such that $xy=n$, then number of ordered pairs $\left(x,y\right)$ where $x,y\in N$ is

If number of 5-digit number of the form $abcde$ where, $a,b,c,d,e\in \left\{0,1,2,...,9\right\}$ and $b=a+c$, $d=c+e$ is $\lambda$, then $\lambda$ is

The number of $3$ digit odd numbers, that can be formed by using the digits $1,2,3,4,5,6$ when the repetition is allowed, is

The number of ways in which $7$ pencils, $6$ books and $5$ pens be disposed off is

$n$ dice are rolled. The number of possible outcomes for which at least one of the dice shows an even number is $189,$then $n=$

The digit in the unit's place of ${\left(141414\right)}^{12121}$ is

The number of $3$-digit odd numbers divisible by $3$ that can be formed using the digits $1,2,3,4,5,6$ when repetition is not allowed is

The number of integers greater than $6000$ that can be formed with $3, 5, 6, 7$ and $9$ where no digit is repeated, is

A person can go from place '$A$' to '$B$' by $11$ different modes of transport but is allowed to return back to "$A$" by any mode other than the one earlier. The number of different ways, the entire journey can be complete is

In a certain test, $a_i$ students gave wrong answers to at least $i$ questions, where $i=1,2,\dots ,k$. No student gave more than $k$ wrong answers. The total number of wrong answers given is equal to

A group of $5$ students, $X,Y,Z,P,Q,$ have the following combination of two subjects each in their first year of college.
$X$ studies Physics and Mathematics, $Y$ studies Physics and Biology, $Z$ studies Biology and Mathematics, $P$ studies Chemistry and Physics and $Q$ studies Mathematics and Chemistry.
A sixth student $R$ joins this group. If he has a subject in common with each of the other students, then his possible subjects of study are

Five friends have the following hobbies :
Akshay likes singing and cooking, Bharati likes singing and painting, Charu likes painting and cooking, Disha likes gardening and singing and Farah likes cooking and gardening. The two hobbies (amongst singing, cooking, painting and gardening), which are least popular with these friends are

Ameeya has $25$ cards, each having a different integer from $1$ to $25$ printed on it. He wishes to place $N$ cards in a single row so that the numbers on every pair of adjacent cards have a prime factor in common. The largest value of $N$ for which this is possible is

There are $6$ multiple choice questions in an examination. How many sequences of answers are possible, if the first three questions have $4$ choices each and the next three have $2$ each?

Suppose a bakery has a selection of $20$ different cupcakes, $10$ different donuts, and $15$ different muffins. If you are to select a tasty treat, how many different choices of sweets can you choose from?