Embibe Experts Solutions for Chapter: Permutation and Combination, Exercise 1: JEE Advanced Paper 2 - 2018

Let $S_1=\left\{\left(i,j,k\right) :i,j,k\in \left\{1,2,\dots ,10\right\}\right\}$

$S_2=\left\{\left(i,j\right) :1\le i<j+2\le 10,i,j\in \left\{1,2,\dots ,10\right\}\right\}$

$S_3=\left\{\left(i,j,k,l\right):1\le i<j<k<l,i,j,k,l\in \left\{1,2,\mathrm{.....},10\right\}\right\}$

and  $S_4=\left\{\left(i,j,k,l\right) : i,j,k \text{and} l \text{are distint elements in} \left\{1,2,...,10\right\}\right\}$

If the total number of elements in the set $S_r$ is $n_r,r=1,2,3,4$ then which of the following statements is (are) TRUE?

An engineer is required to visit a factory for exactly four days during the first $15$ days of every month and it is mandatory that no two visits take place on consecutive days. Then the number of all possible ways in which such visits to the factory can be made by the engineer during $1-15$ June $2021$ is_____

In a hotel, four rooms are available. Six persons are to be accommodated in these four rooms in such a way that each of these rooms contains at least one person and at most two persons. Then the number of all possible ways in which this can be done is_____

Suppose $\det \left[\begin{array}{cc}\sum _{k=0}^{n}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{k}^2\\ \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_{k}k& \sum _{k=0}^n{}_{\mathrm{ }}{}^n{C}_k{3}^k\end{array}\right]=0,$ holds for some positive integer $n.$ Then $\sum _{k=0}^n\frac{{}_{ }{}^n{C}_k}{k+1}$ equals

Five persons $\mathrm{A},\mathrm{ }\mathrm{B},\mathrm{ }\mathrm{C},\mathrm{ }\mathrm{D}$ and $\mathrm{E}$ are seated in a circular arrangement. If each of them is given a hat of one of the three colours red, blue and green, then the number of ways of distributing the hats such that the persons seated in adjacent seats get different coloured hats is

Let $X$ be a set with exactly $5$ elements and $Y$ be a set with exactly $7$ elements. If $\alpha$ is the number of one-one functions from $X\to Y$ and $\beta$ is the number of onto functions from $Y\to X$, then the value of $\frac{1}{5!}\left(\beta -\alpha \right)$ is _____-.

Let $S=\left\{\mathrm{1,2},3,\dots .9\right\}$. For $k=\mathrm{1,2},\dots 5,$ let $N_k$ be the number of subsets of $S$, each containing five elements out of which exactly $k$ are odd. Then $N_1+N_2+N_3+N_4+N_5=$

The number of $5$ digit numbers which are divisible by $4$, with digits from the set $\{\mathrm{1,2},\mathrm{3,4},5\}$ and the repetition of digits is allowed, is _____.