Permutation and combination practice · Medium

There are ' $n$ ' numbered seats around a round table. Total num- ber of ways in which $n_{1}\left(n_{1} < n\right)$ persons can sit around the round table is equal to

$n$ - digit numbers are formed using only three digits 2,5 and 7. The smallest value of $n$ for which 900 such distinct numbers can be formed, is

Total number of four digit odd number that can be formed using $0,1,2,3,5,7$ and having distinct digits are

There are $(n+1)$ white similar balls and $(n+1)$ black balls of different size. No. of ways the balls can be arranged in a row so that adjacent balls are of different colours is

In a club election, the number of contestants is one more than the number of maximum candidates for which a voter can vote. If the total number of ways in which a voter can vote be 62, then the number of candidates is

If the number of ways of selecting $K$ coupons out of an unlimited number of coupons bearing the letters A, T, M, so that they cannot be used to spell the word MAT is 93, then $K$ is equal to

The permutations of $n$ objects taken

(i) atleast $r$ objects at a time

(ii) atmost $r$ objects at a time
(if repetition of the objects is allowed) are respectively

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

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

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

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

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

$n$ 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

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

Nine hundred distinct N –digit numbers are to be formed by using 6, 8 and 9 only. The smallest
value of N for which this is possible is

Nine toys are to be packed in 9 boxes. If 5 of them are too big for 3 boxes, then the number of ways in which they can be packed is

The number of ways in which the letters of the word VOWEL can be arranged so that the letters O, E occupy only even places, is

${ }^{4 n} C_{2 n}:{ }^{2 n} C_n$ is equal to

Two teachers are taking 6 students to a zoo. The teachers decide to split up. Each student must choose one of the teachers, with the condition that each teacher must take at least one student. Number of possible ways of doing this is:

Let $a_n=$ number of $n$-digit positive integers formed by the digits 0, 1 or both such that no consecutive digits in them is 0.
Let $b_n=$ number of such $n$ digit integers ending with digit 1, and $c_n=$ the numner of such $n$ digit integers ending with digit 0, then for $n \geq 3$,

The number of ways in which the digits $1,2,3,4,3$, 2,1 can be arranged so that the odd digits always occupy the odd places, is

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

If two numbers are selected from numbers 1 to 25, then the number of ways that their difference does not exceed 10 is

Six boys and six girls sit along a line alternately in $x$ ways; and along a circle again alternatively in $y$ ways, then

In a certain test, there are $n$ questions. In this test, $2^{n- i}$ students gave wrong answers to atleast $i$ questions, where $i =1, 2, 3,..., n.$ If the total number of wrong answers given is 2047, then $n$ is equal to

The number of permutations of all the letters of the word 'EXERCISES' is