Permutation and combination practice · Mixed difficulty

The sum of all 4 digited numbers that can be formed using the digits $1,2,4,5,6$ without repetition is

If $N$ is the number of positive integral solution of ${x_1}{x_2}{x_3}{x_4} = 770$, then the value of $N$ is

If ${\;^{12}}{P_r} = {\;^{11}}{P_6} + 6.{\;^{11}}{P_5}$ then $r$ is equal to

If seven women and seven men are to be seated around a circular table such that there is a man on either side of every woman, then the number of seating arrangements is

There are 3 set of parallel lines containing respectively $p$ lines, $q$ lines and $r$ lines respectively. The greatest number of parallelograms that can be formed by the system

The number of lines drawn through 6 points lying on a circle is

How many 10 digits numbers can be written by using digits 9 and 2?

The exponent of 12 in 50! is

Total number of three letter words that can be formed from the letters of the word 'SAHARANPUR' is equal to

The number of natural numbers which are less than $2 \cdot 10^8$ and which can be written by means of the digit 1 and 2, is

In a circus there are ten cages for accommodating ten animals. Out of these four cages are so small that five out of 10 animals cannot enter into them. In how many ways will it be possible to accommodate ten animals in these ten cages?

If the permutations of $a,b,c,d,e$ taken all together be written down in alphabetical order as in dictionary and numbered, then the rank of the permutation $debac$ is

If ${ }^{2 n+1} P_{n-1}:{ }^{2 n-1} P_n=3: 5$, then $n$ is equal to

Let $p$ be a prime number such that $p \geq 3$. Let $n=p !+1.$ The number of primes in the list $n + 1, n + 2, n + 3, ...., n + p − 1$ is

How many numbers divisible by 5 and lying between 3000 and 4000 that can be formed from the digits 1, 2, 3, 4, 5, 6 (repetition of digits is not allowed)?

$\lim\limits_{n \to \infty } \frac{{\sum\limits_{1 \leqslant i} {\sum\limits_{ \leqslant j \leqslant n} {\left( {i + 1} \right)\left( {j + 1} \right)} } }}{{{n^4}}}$ is equal to

If $[y]$ denote the greatest integer $\leq y$, and $2\left[\frac{x}{8}\right]^2+3\left[\frac{x}{8}\right]=20$, then $x$ lies in the smallest interval $[a, b)$ where $b-a$ is equal to

There are 20 questions in a question paper. If no two students solve the same combination of questions but solve equal number of questions, then the maximum number of students who appeared in the examination, is

How many different arrangements can be made out of the letters in the expansion $A^2 B^3 C^4$, when written in full?

If ${\;^n}{P_r} = 30240$ and ${\;^n}{C_r} = 252$, then the ordered pair $\left( {n,\;r} \right)$ is equal to

In an examination a minimum is to be secured in each of 5 subjects for a pass. The number of ways in which a student can fail is

$A_1, A_2 \ldots A_n$ are points on the line $y=x$ lying in the positive quadrant such that $O A_n=n O A_{n-1}, O$ being the origin. If $O A_1=1$ and the coordinates of $A_n$ are $(2520 \sqrt{2}, 2520 \sqrt{2})$, then $n=$

Let ${P_m}$ stand for ${\;^m}{P_m}.$ Then, $1 + {P_1} + 2\;{P_2} + 3\;{P_3} + \ldots + n.{P_n}$ is equal to

The number of ways in which the letters of the word "STRANGE" can be arranged so that the vowels may appear in the odd places, is

The number of ways of selecting $n$ objects out of $3 n$ objects, of which $n$ are alike and rest are different, is

The number of all possible words that can be formed using the letters of the word “MATHEMATICS” is

Find the number of ways of arranging the letters of the word 'BANANA'

The number of ways in which 20 different pearls of two colours can be set alternately on a necklace, there being 10 pearls of each colour, is

The total number of ways of arranging the letters AAAA BBB CC D E F in a row such that letters C are separated from one another is

Out of 7 consonants and 4 vowels, the number of words (not necessarily meaningful) that can be made, each consisting of 3 consonants and 2 vowels, is