Permutation and combination practice · Easy

How many numbers with no more than three digits can be formed using only the digits 1 through 7 with no digit used more than once in a given number?

The number of integral solutions of $x_1+x_2+x_3=0$, with $x_i \geq-5$, is

The number of different pairs of words $(\square \square \square \square, \square \square \square)$ that can be made with the letters of the word 'STATICS' is

In how many ways can 5 boys and 5 girls sit in a circle so that no two boys sit together?

The largest integer $n$ such that 33 ! is divisible by $2^n$ is

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

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

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

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 $x+y$.

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:

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

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 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

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

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

If ${P_m}$ stands for ${\;^m}{P_m}$, then $1 + 1{P_1} + 2{P_2} + 3{P_3} + ... + n.{P_n}$ is equal to

$2 m$ white counters and $2 n$ red counters are arranged in a straight line with $(m+n)$ counters on each side of a central mark. The number of ways of arranging the counters, so that the arrangements are symmetrical with respect to the central mark, is

Number of ways of selection of 4 letters of the word "PERFECT" is

The value of ${2^n}\left[ {1.3.5 \ldots \left( {2n - 3} \right)\left( {2n - 1} \right)} \right]$ is

There are stalls for 10 animals in a ship. The number of ways the shipload can be made if there are cows, calves and horses to be transported, animals of each kind being not less than 10 , is

The number of ways of arranging $m$ numbers out of $1,2,3, \ldots, n$, so that maximum is $(n-2)$ and minimum is 2 (repetitions of numbers is allowed) such that maximum and minimum both occur exactly once $(n > 5, m > 3)$, is

The number of ways in which 10 candidates $A_1, A_2, \ldots$, $A_{10}$ can be ranked so that $A_1$ is always above $A_2$, is