Permutation and combination practice · Mixed difficulty

The total number of natural numbers of six digits that can be made with digits 1,2,3,4, if all digits are to appear in the same number at least once, is

The number of numbers less than 1000 that can be formed out of the digits $0,1,2,3,4$ and 5 , no digit being repeated, is

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

The number of all four digit numbers which are divisible by 4 that can be formed from the digits 1,2,3,4 and 5 is

The number of ways in which the letters of the word 'SUCCESSFUL' be arranged such that the S's and U's will come together 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

The number of numbers, strictly between 5000 and 10000 can be formed using the digits $1,3,5,7,9$ without repetition, is

There are 20 students in a class, in which 12 boys and 8 girls. The class teacher select either a boy or a girl for monitor of the class. In how many ways, the teacher can make this selection?

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 9 digit numbers which have all distinct digits, 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:

The last digit of $(1 !+2 !+\ldots \ldots+2009 !)^{500}$ is

In how many ways the letters of the word ENGINEERING, can be arranged such that all E's come together but no two N's come together

The number of arrangements that can be formed out of 'LOGARITHM' so that "No two vowels come together is

If ${ }^{{m}+{n}} P_2=90$ and ${ }^{{m}-{n}} P_2=30$, then $(m, n)$ is equal to

The number of different matrices that can be formed with elements $0,1,2$ or $3,$ each matrix having 4 elements, 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

How many numbers lying between 10 and 1000 can be formed from the digits 1, 2, 3, 4, 5, 6, 7, 8, 9 (repetition of digits is allowed)?

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

The number of arrangements that can be formed out of GANESHPURI so that "The letter 'G' always occurs in first place is

If the letters of the word MOTHER are written in all possible orders and these words are written out as in a dictionary then the rank of the word MOTHER is

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

The number of ways of choosing triplets $(x, y, z)$ such that $z \geq \max \{x, y\}$ and $x, y, z \in\{1,2, \ldots, n, n+1\}$, is

The sum of all four digit numbers that can be formed by using digits $2,4,6,8$ (when repetition of digits is not allowed) is