Permutation and combination practice · Mixed difficulty

A man invites 6 non-vegeterian and 5 vegeterian friends for a dinner party. He arrange 6 non-vegeterian friends on one round table and 5 vegeterian friends along another round table. The number of ways this can be done is:

Ten different letters of an alphabet are given. Words with five letters are formed from these given letters. Then the number of words which have at least one letter repeated, is

The least positive integral value of $n$ which satisfies the inequality ${ }^{10} {C}_{n-1}>2 \cdot{ }^{10} {C}_n$ is

The value of $\left(2 \cdot{ }^{1} P_{0}-3 \cdot{ }^{2} P_{1}+4 \cdot{ }^{3} P_{2}-\ldots\right.$ up to $51^{\text {th }}$ term $)$ $+\left(1 !-2 !+3 !-\ldots\right.$ up to $51^{\text {th }}$ term $)$ is equal to :

The number of arrangements of the letters of the word BANANA in which the two N's do not appear adjacently, is

The total number of ways in which five '$+$' and three '$-$' signs can be arranged in a line such that no two '$-$' sign occur together, is

The rank of the word MOTHER when the letters of the word are arranged alphabetically as in a dictionary, is

Out of thirty points in a plane, eight of them are collinear. The number of straight lines that can be formed by joining these points, is

A set contains $(2 n+1)$ elements. The number of subsets of the set which contain at most $n$ elements is

If the letters of the word KRISNA are arranged in all possible ways and these words are written out as in a dictionary, then the rank of the word KRISNA is

Let $A$ be the set of 4-digit numbers $a_1 a_2 a_3 a_4$, where $a_1 > a_2 > a_3 > a_4$, then $n(A)$ is equal to

How many numbers lying between 999 and 10000 can be formed with the help of the digits 0, 2, 3, 6, 7, 8 when the digits are not be repeated?

If each of 10 points on a straight line be joined to each of 10 points on a parallel line then the total number of triangles that can be formed with the given points as vertices, is

The value of ${ }^{50} C_4+\sum\limits_{{r=1}}^6{ }^{56-r} C_3$ is

Number of 4 digit numbers of the form $N=a b c d$ which satisfy following three conditions:

(i) $4000 \leq N < 6000$

(ii) $N$ is a multiple of 5

(iii) $3 \leq b < c \leq 6$

is equal to:

The number of 6-digit numbers that can be made with the digits $1,2,3$ and $4$ and having exactly two pairs of digits is

There are $n$ different books and $p$ copies of each. The number of ways in which a selection can be made from them is

If $x \in {N}$ and $\frac{(x+2) !}{(2 x-1) !} \cdot \frac{(2 x+1) !}{(x+3) !}=\frac{72}{7}$, then $x$ is equal to

The number of natural numbers smaller than ${10^4},$ in the decimal notation of which all the digit are different is

The integers from 1 to 1000 are written in order around a circle. Starting at 1, every fifteenth number is marked (i.e. 1, 16, 31 etc.) This process is continued until a number is reached which has already been marked, then unmarked numbers are

There are $p$ copies each of $n$ different books. The number of different ways in which a non-empty selection can be made from them, is

The number of ways in which three girls and nine boys can be seated in two cars, each having numbered seats, 3 in the front and 4 at the back, is

There are 5 historical moments, 6 gardens and 7 shopping malls in the city. In how many ways a tourist can visit the city, if he visits at least one shopping mall?

The total number of ways in which 4 boys and 4 girls can form a line, with boys and girls alternating, is

The number of three digit numbers of the form $x y z$ such that $x < y,\,\,\,z \leqslant y$ and $x \ne 0$, is

Let $n_1=x_1 x_2 x_3$ and $n_2=y_1 y_2 y_3$ be two 3-digit numbers, then the pairs of $n_1$ and $n_2$ can be formed, so that $n_1$ can be subtracted from $n_2$ without borrowing, is

Nine hundred distinct $\mathrm{n}$-digit number $\mathrm{s}$ are to be forme using exactly the three digits. 2, 5 and 7 . The smallest value&nbsp; of $\mathrm{n}$ for which this is possible is.

A gentleman invites 13 guests to a dinner and places 8 of them at one table and remaining 5 at the other, the tables being round. The number of ways he can arrange the guests is

The sum of the digits in the unit place of all numbers formed with the help of 3, 4, 5, 6 taken al, at a time, is

$n$ is selected from the set $\{1,2,3, \ldots, 100\}$ and the number $2^n+3^n+5^n$ is formed. Total number of ways of selecting $n$, so that the formed number is divisible by 4, is equal to