Embibe Experts Solutions for Chapter: Permutation and Combination, Exercise 3: Level 3

If the sum of all numbers greater than $10000$ formed by using the digits $0,1,2,4,5$ & no digit being repeated in any number, is $N$, then sum of digits of $N$ is

The number of different seven digit numbers that can be written using only three digits $1, 2$ & $3$ under the condition that the digit $2$ occurs exactly twice in each number is -

If there are $N$ integers between $100$ and $999$, inclusive, have the property that some permutation of its digits is a multiple of $11$ between $100$ and $999?$ For example, both $121$ and $231$ have this property. Then find the value of $\frac{N-1}{5}.$

Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different.)

Let $a_1, a_2, a_3, \dots$ be the sequence of all positive integers that are relatively prime to $75$, where $a_1<a_2<a_3<\cdots .$ (The first five terms of the sequence are: $a_1=1, a_2=2, a_3=4, a_4=7, a_5=8.$)
Then the value of $a_{2008}$ is defined as $p·{10}^3+q·{10}^2+r·10+s$. Find $\left(p+q+r+s\right)$

All diagonals are drawn in a regular octagon. At how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?

If from the set $1, 2, 3, ..., 2017$, the number of ways in which one can choose two different numbers whose sum is an even number be $n$, then $n-1016000$ is

If there are $5$ letters written to $5$ different people and $5$ envelopes addressed to them, then the number of ways in which these letters can be arranged so that no letter goes into its corresponding envelope is