Fundamental Principle of Counting

If $15$ people won prizes in the state lottery (assuming that there are no ties), how many ways can these $15$ people win first, second, third, fourth and fifth prizes?

In each question below a combination of name and address is given, followed by four such options, one each under the columns A,B, C and D. You have to find out the combination which is exactly the same as the combination in the question. The number of that option which contains that combination is the answer.

Yogi chaphriderNiranjan AshramNaseek -422013

How many permutations can be made out of the letters in the word island taking four letters at a time?

Five different Mathematics books,$4$ different electronics books, and $2$different communications books are to be placed on a shelf with the books of the same subject together. Find the number of ways in which the books can be placed?

A motorist knows four different routes from Bristol to Birmingham. From Birmingham to Sheffield he knows three different routes and from Sheffield to Carlisle he knows two different routes. How many routes does he know from Bristol to Carlisle ?

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 ?

In how many ways can $4$ boys and $4$ girls be seated alternately in a row of $8$ seats?

What is the number of $6$-digit numbers that can be formed only by using $0, 1, 2, 3, 4$ and $5$ (each once); and divisible by $6$?

$5$-digit numbers are formed using the digits $0, 1, 2, 4, 5$ without repetition. What is the percentage of numbers which are greater than $50,000$?

Let $x$ be the number of permutations of the word 'PERMUTATIONS' and $y$ be the number of permutations of the word 'COMBINATIONS'. Which one of the following is correct?

Total numbers of $3$-digit numbers that are divisible by $6$ and can be formed by using the digits $1, 2, 3, 4, 5$ with repetition, is ________

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

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)

Let the number of matrices of order $3\times 3$ are possible using the digits $\left\{0,2,3,...,10\right\}$ is $m^n$, then $m+n$ is

$N>40000$, where $N$ is divisible by $5$. How many such $5$ digit numbers can be formed using $0,1,3,5,7,9$ without repition.

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)

One has $10$ different shirts, $5$ different trousers, $5$ different ties, $3$ different pair of shoes, In how many different ways can he wear them if wearing all the items necessary

Show that if $6$ integers are selected from first $10$ positive integers, there must be a pair of these integers with sum $11$.