Permutation and combination practice · Mixed difficulty

Let $A$ be a set of $n(\geq 3)$ distinct elements. The number of triplets $(x, y, z)$ of the set $A$ in which atleast two coordinates are equal to

Out of 18 points in a plane no three are in the same straight line except five points which are collinear. The number of straight lines that can be formed joining them is

Let $T_n$ be the number of all possible triangles formed by joining vertices of an $n$-sided regular polygon. If $T_{n+1}-T_n=10$, the value of $n$ is

A seven-digit number made up of all distinct digits $8,7,6,4,2, x$ and $y$ is divisible by 3. Then, possible number of order pair $(x, y)$ is

If $r,\;s,\;t$ are prime numbers and $p,q$ are the positive integers such that LCM of $p,\;q$ is ${r^2}{s^4}{t^2}$, then the number of ordered pairs $\left( {p,\;q} \right)$ is

How many numbers greater than 40000 can be formed from the digits 2, 4, 5, 5, 7?

How many words can be made from the letters of the word ‘COMMITTEE’?

Every one of the 10 available lamps can be switched on to illuminate certain Hall. The total number of ways in which the hall can be illuminated, is

Sixteen men compete with one another in running swimming and riding. How many prize lists could be made if there were altogether 6 prizes of different values, one for running, 2 for swimming and 3 for riding?

A person predicts the outcome of 20 cricket matches of his home team. Each match can result either in a win, loss or tie for the home team. Total number of ways in which he can make the predictions, so that exactly 10 predictions are correct, is equal to

A four digit number of distinct digits is formed by using the digits $2,3,4,5,6,7,8$. The number of such numbers which are divisible by 25, is

The least positive integer $n$ for which ${ }^{n-1} C_5+{ }^{n-1} C_6 < { }^n C_7$ is

The exponent of 3 in 100 !, is

Number of ways is which 50 identical books can be distributed in 100 students if each student can get atmost one book is

There are 10 true-false questions. The number of ways, in which they can be answered is

The value of ${ }^{10} C_5 /{ }^{11} C_6$, when numerator and denominator takes their greatest value, is

If all permutations of the letters of the word PENCIL are arranged as in a dictionary, then 413th word is

A seven-digit numbers divisible by 9 is to be formed by using 7 out of numbers $\{1,2,3,4,5, 6,7,8,9\}$. The number of ways in which this can be done, is

How many four digit numbers can be formed using the digits 1, 2, 3, 4, 5 such that at least one of the digit is repeated?

A dictionary is printed consisting of 7 letters words only that can be made with a letters of the word CRICKET. If the words are printed at the alphabetical order as in an ordinary dictionary, then the number of words before the word CRICKET is

The number of six-digit numbers that can be formed from the digits 1, 2, 3, 4, 5, 6 and 7, so that digits do not repeat and the terminal digits are even, is

The number of words of four letters containing equal number of vowels and consonants, repetition allowed, is

The number of words which can be formed the letters of the word MAXIMUM, if two consonants cannot occur together, is

The number of ways in which $p$ positive and $n$ negative signs may be placed in a row so that no two negative signs shall be together is

The number of 6 digits numbers that can be formed using the digits 0, 1, 2, 5, 7 and 9 which are divisible by 11 and no digit is repeated, is

Let $S$ be the set of all functions from the set $A$ to the set $A$. If $n(A)=k$, then $n(S)$ is

If ${ }^{10} C_{{r}-1} > 2\left({ }^{10} C_{\mathrm{r}}\right)$ then $r$ is equal to

If $x \in {N}$ and $\frac{(2 x) !}{3 !(2 x-3) !}: \frac{x !}{2 !(x-2)}=44: 3$, then $x$ is equal to

A basket ball team consists of 12 pairs of twin brothers. On the first day of training, all 24 players stand in a circle in such a way that all pairs of twin brothers are neighbours. Number of ways this can be done is :

A variable name in certain computer language must be either an alphabet or an alphabet followed by a decimal digit. The total number of different variable names that can exist in that language is equal to