Permutation and combination practice · Mixed difficulty

The number of all five digit numbers which are divisible by 4 that can be formed from the digits 0,1,2,3,4 (without repetition) is

In how many ways can 10 examination paper be arranged, so that the best and the worst papers never come together?

The numbers of times the digits 3 will be written when listing the integers from 1 to 1000 is

Four boys picked 30 apples. The number of ways in which they can divide them if all the apples are identical, is

The total number of ways can 15 identical Mathematics books be distributed among six students, is

The number of natural numbers less than 1000, in which no two digits are replaced, is

No. of ways to give 16 different things to $A, B, C$ so that $B$ gets 1 more than $A$ and $C$ gets 2 more than $B$, is

The number of ways such that 6 persons can be selected from 4 officers and 8 constables, if at least one officer is to be included, is

Three dice are rolled. The number of possible outcomes in which at least one die shows 5 is

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

When a group photograph is taken, all the seven teachers should be in the first row and all the twenty students should be in the second row. If the two corners of the second row are reserved for the two tallest students, interchangeable only between them, and if the middle seat of the front row is reserved for the principal. The no. of such possible 'arrangements' is

In a plane, there are two sets of parallel lines, one of $m$ lines and the other of $n$ lines. If the lines of one set cut those of the other, then the number of different parallelograms that can be formed is

$m$ men and $w$ women are to be seated in a row so that all women sit together. The number of ways in which they can be seated is

$n$ - 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

If all the words with or without meaning made using all the letters of the word "NAGPUR" are arranged as in a dictionary, then the word at $315^{\text {th }}$ position in this arrangement is :

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

The number of natural numbers less than one million that can be formed by using the digits, 0, 2, and 3 is

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

There are 4 candidates for the post of a lecturer in Mathematics and one is to be selected by votes of 5 men. The number of ways in which the votes can be given is

If there are 62 onto mapping from a set $X$ containing $n$ elements to the set $Y=\{-1,1\}$, then $n$ is equal to:

From 6 different novels and 3 different dictionaries, 4 novels and 1 dictionary are to be selected and arranged in a row on a shelf, so that the dictionary is always in the middle. The number of such arrangements is

There are ' $n$ ' seats round a table numbered $1,2,3, \ldots n$ The numbers of in which $m(\leq n)$ persons can take seats 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

The number of ways in which the letters of the word 'SUCCESSFUL' be arranged such that all the ' $s$ ' come together 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

In how many ways 4 alphabets can be chosen from the word ASSASSINATION?

The number of ways in which 7 people can be arranged at a round table, so that 2 particular persons may be together, is

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

From 17 consonants and 5 vowels. The number of words containing 3 consonants and 2 vowels that can be made if all the letters are different, is

Let $x_1, x_2, x_3, \ldots, x_k$ be the divisors of positive integer ' $n$ ' (including 1 and $x$ ). If $x_1+x_2+\ldots+x_k=75$, then $\sum\limits_{i=1}^k \frac{1}{x_i}$ is equal to