Permutation and combination practice · Mixed difficulty

If $n=2^{p-1}\left(2^p-1\right)$, where $2^p-1$ is a prime, then the sum of the divisors of $n$ is equal to

$\sum \limits_{r = 0}^m {\;^{n + r}}{C_n}$ is equal to

The number of permutations of 4 letters that can be made out of the letters of the word EXAMINATION is

The number of arrangements of the letters of the word "pencil" so that $e$ always comes earlier than $i$, is

Out of 5 apples, 10 mangoes and 15 oranges, the number of ways of distributing 15 fruits each to two persons, is $\mathrm{n}$ then $\frac{\mathrm{n}}{22}$ is equal to

Two players $P_1$ and $P_2$ play a series of '$2 n$' games. Each game can result in either a win or loss for $P_1$. Total number of ways in which $P_1$ can win the series of these games, is equal to

Let $f:\left\{ {1,\;2,\;3,\;4,\;5} \right\} \to \left\{ {1,\;2,\;3,\;4,\;4,\;5} \right\}$ that are onto and $f\left( x \right) \ne i$ is equal to

15 identical balls have to be put in 5 different boxes. Each box can contain any number of balls. Total number of ways of putting the balls into box, so that each box contains atleast 2 balls, is equal to

The number of numbers lying between 40000 and 70000 that can be made from the digits $0,1,2,4,5,7$, if digits can be repeated in the same number, is

number of ways in which 15 indistinguishable oranges can be distributed in 3 different boxes so that every has atmost 8 oranges is

Total number of 6 digit numbers in which all the odd digits and only odd digits appear is

How many four-digit numbers can be formed using the digits $0,1,2,3,4,5$, if repetition of digits is not allowed?

The number of ordered pairs $(m, n)$, $m, n \in\{1,2, \ldots, 100\}$ such that $7^m+7^n$ is divisible by 5, is

The number of word formed by using all letter of word 'RANDOM' in which vowels are in alphabetical order is

All letters of the word EAMCET are arranged in all possible ways. The number of such arrangement in which no two vowels are adjacent to each other, is

In a certain test, $a_i$ students gave wrong answers to atleast $i$ questions, where $i=1,2,3, \ldots, k$. No student gave more than $k$ wrong answers. The total number of wrong answers given is

Find the number of different necklaces that can be made from 17 identical pearls and 2 different diamonds.

Let $S=\{1,2,3, \ldots, n\}$. If $X$ denotes the set of all subsets of $S$ containing exactly two elements, then the value of $\sum\limits_{{A \in X}}(\min A)$ is

Five distinct letters are to be transmitted through a communication channel. A total number of 15 blanks is to be inserted between the two letters with at least three between every two. The number of ways in which this can be done is $t$ then $\frac{t}{400}$ is.

The sum of factors of $8 !$ which are odd and are of the form $3 m+2$, where $m$ is a natural number, is

The letters of the word MODESTY are written in all possible orders and these words are written out as in a dictionary, then the rank of the word MODESTY is

In a single correct match the column question, column I contain 10 questions and Column II contain 10 answers written in some arbitrary order. If the number ways a student can answer this question so that exactly 6 of his matching are correct is $\mathrm{k}$, then (sum of digits of k) $/ 2$ is equal to

Let a be a factor of 120 . Then the number of positive integral solutions of $\mathrm{x}_1, \mathrm{x}_2, \mathrm{x}_3=\mathrm{a}$ is $\mathrm{k}$ then $\frac{\mathrm{k}}{80}$ is

The number of ways in which two 10 paise, two 20 paise, three 25 paise and one 50 paise coins can be distributed among 8 children, so that each child gets only one coin, is

If ${ }^{n+5} p_{n+1}=\frac{1}{2}(11)(n-1)\left({ }^{n+3} p_n\right)$ then $n$ is equal to:

The number of ways in which the sum of upper faces of four distinct dices can be six, is

The number of arrangements that can be formed out of GANESHPURI so that "The vowels occupy even places are

The number of ways in which 3 children can distribute 10 tickets out of 15 consecutively numbered tickets themselves such that they get consecutive blocks of 5, 3 and 2 tickets is

In the decimal system of numeration, the number of 6-digit numbers in which the sum of digits is divisible by 5, is

How many numbers between 5000 and 10,000 can be formed using the digits 1,2,3,4,5,6,7,8,9, each digit appearing not more than once in each number?