Permutation and combination practice · Mixed difficulty

The positive integer $r$, such that ${ }^{15} {C}_{3 r}={ }^{15} {C}_{r+3}$, is equal to

The value of $\sum\limits_{{r=1}}^{15} r^2\left(\frac{{ }^{15} C_r}{{ }^{15} C_{r-1}}\right)$ is equal to:

If the number of words, with or without meaning, which can be made using all the letters of the word MATHEMATICS in which C and S do not come together, is $(6 !) k$, then $k$ is equal to

The number of numbers of 4 digits which are not divisible by 5, are

The number of 5-digit numbers that can be made using the digits 1 and 2 and in which atleast one digit is different, is

ABCD is a quadrilateral. $3,4,5$ and 6 points are marked on the sides $\mathrm{AB}, {BC}, {CD}$ and ${DA}$ respectively. The number of triangles with vertices on different sides is

Let $Q_{{n}}$ be the number of possible quadrilaterals formed by joining vertices of an $n$ sided regular polygon. If $Q_{n+1}-Q_n=20$, then value of $n$ is:

The number of permutations by taking all letters and keeping the vowels of the word COMBINE in the odd places is

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 by joining them is

How many words can be formed from the letters of the word DOGMATIC, if all the vowels remain together?

$\frac{(2 n) !}{2^{2 n}(n !)^2} \leq$

The number of all possible selections of one or more questions from 10 given questions, each question having an alternative is

All possible four-digit numbers are formed using the digits 0,1,2,3 so that no number has repeated digits. The number of even number among them is

In how many ways can 21 English and 19 Hindi books be placed in a row so that no two Hindi books are together?

${\;^n}{P_r} = 3024$ and ${\;^n}{C_r} = 126$, then $r$ is

The number of numbers lying in $(2,3)$, where all the digits after decimal are non-zero and in increasing order, is

If $33 !$ is divisible by $2^n$, then the maximum value of $n$ is equal to

12 persons are to be arranged to a round table. If two particular persons among them are not to be side by side, the total number of arrangements is

The number of rectangles excluding squares from a rectangle of size $9 \times 6$ is

If eleven member of a committee sit at a round table so that the President and Secretary always sit together, then the number of arrangements is

The total numbers of greater than 100 and divisible by 5, that can be formed from the digits 3,4,5,6 if no digit is repeated is

Find the number of permutations $x_{1}, x_{2}, x_{3}, x_{4}, x_{5}$ of numbers $1,2,3,4,5$ such that the sum of five products $x_{1} x_{2} x_{3}+x_{2} x_{3} x_{4}+x_{3} x_{4} x_{5}+x_{4} x_{5} x_{1}+x_{5} x_{1} x_{2}$ is divisible by 3

Garlands are formed using 6 red roses and 6 yellow roses of different sizes. The number of arrangements in garland which have red roses and yellow roses come alternately is

From 4 Officers and 8 Jawans, a committee of 6 is to be chosen to include exactly one officer. The number of such committee is

The number of ways in which ten candidates $A_1, A_2, A_3, A_4, \cdots, A_{10}$ can be arranged in a row "If $A_1$ is always above $A_2$ is

A teaparty is arranged for 16 people along two sides of a large table with 8 chairs on each side. Four men want to sit on one particular side and two on the other side. The number of ways in which they can be seated, is

There are three teams each of a chairman, a supervisor and a worker three different companies. There are nine bonus of different denominations to be paid to these nine persons in all. In how many ways can this be done with due respect to superiority is given in every team?

In a shooting competition, a man can score 5, 4, 3, 2 or 0 points for each shot. Then, the number of different ways in which he can score 30 in seven shots, is

The number of possible outcomes in a throw of $n$ ordinary dice in which atleast one of the dice shows an odd number, is

In a class, there are 15 boys and 10 girls. The teacher wants to select a boy and a girl to represent the class in a function, then the total number of selection by teacher is