Permutation and combination practice · Mixed difficulty

If $m_1$ and $m_2$ satisfy the relation ${ }^{m+5} P_{m+1}=\frac{11}{2}(m-1)\left({ }^{m+3} P_m\right),$ then $m_1+m_2$ is equal to

If eight persons are to address a meeting, then the number of ways in which a specified speaker is to speak before another specified speaker is

The number of ways in which the letters of the word ARRANGE be arranged so that "two A's are together but not two R's

A bag contains 2 apples, 3 oranges and 4 bananas. The number of ways in which 3 fruits can be selected, if atleast one banana is always in the combination (assume fruit of same species to be alike) is

Total number of non-negative integral solutions of $x_1+x_2+x_3=10$ is equal to

The number of ways of arranging six persons (having $A, B, C$ and $D$ among them) in a row so that $A, B, C$ and $D$ are always in order $A B C D$ (not necessarily together) is

If $\frac{2}{9 !}+\frac{2}{3 ! 7 !}+\frac{1}{5 ! 5 !}=\frac{2^a}{b !}$, where $a, b \in N$, then the ordered pair $(a, b)$ is

The number of ways in which a score of 11 can be made from a through by three persons, each throwing a single die once, is

Words with or without meaning are to be formed using all the letters of the word EXAMINATION. The probability that the letter M appears at the fourth position in any such word is

The number of positive integral solutions of the equation $x_1 x_2 x_3 x_4 x_5=420$ is

The number of proper divisors of 1260 , is

A number of 4 different digits is formed by using the digits $1,2,3,4,5,6,7$ such that number formed is multiple of 25 , is $12 x+y$, then find the value of value of $x+y$.

$\sum\limits_{0 \leqslant i} {\sum\limits_{ \leqslant j \leqslant 10} {^{10}{C_j}^j{C_i}} }$ is equal to

The total number of 3 digit even numbers that can be composed from the digitis $1,2,3, \ldots$. , 9 when the repetition of digits is not allowed, is

There are $n$ concurrent lines and another line parallel to one of them. The number of different triangles that will be formed by the $(n+1)$ lines, is

If the letters of the word "nagpur" are written in all possible ways and then are arranged as in a dictionary, then the rank of the word "nagpur" is:

If $a_n=\sum\limits_{{r=0}}^n \frac{1}{{ }^n C_r}$, then $\sum\limits_{{r=0}}^n \frac{r}{{ }^n C_r}$ equals:

Let $A=\{1,2,3,4,5\}$ and $B=\{1,2,3,4,5\}$. If $f: A \rightarrow B$ is an one-one function and $f(x)=x$ holds only for one value of $x \in\{1,2,3,4,5\}$, then the number of such possible function is

Number of points having position vector $a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$, where $a, b, c \in\{1,2,3,4,5\}$ such that $2^a+3^b+5^c$ is divisible by 4 , is

If number of numbers greater than 3000 , which can be formed by using the digits $0,1,2,3,4,5$ without repetition, is $n$ then $\frac{n}{230}$ is equal to

The number of even divisors of 10800 is

If $a$ denotes the number of permutations of $x+2$ things taken all at a time, $b$ the number of permutations of $x$ things taken 11 at a time and $c$ the number of permutations of $x-11$ things taken all at a time such that $a=182 b c$, the value of $x$ is

A set containing $n$ elements. A subset $P$ of $A$ is chosen. The set $A$ is reconstructed by replacing the element of $P$. A subset $Q$ of $A$ is again chosen the number of ways of choosing $P$ and $Q$, so that $P \cap Q=\phi$, is

The number of words which can be formed out of letters $a, b, c, d, e, f$ taken 3 together, each containing one vowel at least is

The sides $AB, BC, CA$ of a $\Delta ABC$ have 3, 4, 5 interior points, respectively on them. Total number of triangles that can be formed using these points as vertices, is equal to

Ajay writes letters to his five friends and addresses the corresponding. The number of ways can the letters be placed in the envelopes so that at least two of them are in the wrong envelopes is

In word 'ENGINEERING', E's and N's come alternately

If $a, b, c$ are three natural numbers in AP and $a+b+c=21$, then the possible number of values of the ordered triplet $(a, b, c)$ is

If $\frac{1}{8 !}+\frac{1}{9 !}=\frac{x}{10 !}$, then $x$ is equal to

If ${ }^n P_r=720,{ }^n C_r=120$, then $(n, r)$ is