Embibe Experts Solutions for Chapter: Permutation and Combination, Exercise 4: Exercise-4

Prove that $\left(n!\right)!$ is divisible by $n{!}^{\left(n-1\right)!}$

Let $X=\left\{1, 2, 3,\dots \dots , 10\right\}.$ Find the the number of pairs $\left\{A, B\right\}$ such $A\subseteq X.B\subseteq X.A\ne B$ and $A\cap B=\left\{5, 7, 8\right\}$

Consider a $20$ -sided convex polygon $K,$ with vertices $A_1, A_2, \dots , A_{20}$ in that order. Find the number of ways in which three sides of $K$ can be chosen so that every pair among them has at least two sides of $K$ between them. For example, $\left(A_1{A}_2, A_4{A}_5, A_{11}{A}_{12}\right)$ is an admissible triple while $\left(A_1{A}_2, A_4{A}_5, A_{19}{A}_{20}\right)$ is not.

Find the number of $4$-digit numbers (in base $10$ ) having non-zero digits and which are divisible by $4$ but not by $8$.

Find the number of all integer-sided isosceles obtuse-angled triangles with perimeter $2008$.

Let $ABC$ be a triangle. An interior point $P$ of $ABC$ is said to be good if we can find exactly $27$ rays emanating from $P$ intersecting the sides of the triangle $ABC$ such that the triangle is divided by these rays into $27$ smaller triangles of equal area. Determine the number of good points for a given triangle $ABC$

Let $\sigma =\left(a_1, a_2, a_3,\dots , a_n\right)$ be a permutation of $\left(1, 2, 3,\dots , n\right).$ A pair $\left(a_i, a_i\right)$ is said to correspond to an inversion of $\sigma ,$ if $i<j$ but $a_1>a_j,$ (Example : In the permutation $\left(2, 4, 5, 3, 1\right),$ there are $6$ inversions corresponding to the pairs $\left(2, 1\right), \left(4, 3\right), \left(4, 1\right), \left(5, 3\right)\left(5, 1\right), \left(3, 1\right).$ How many permutations of $\left(1, 2, 3,\dots n\right), \left(n\ge 3\right),$ have exactly two inversions.?

Let $A$ is set of all possible planes passing through four vertices of given cube. Find number of ways of selecting four planes from set $A$, which are linearly dependent and one common point. (If planes $P_1=0$, $P_2=0,P_3=0$ and$P_4=0$ can be written as $a{P}_1+b{P}_2+c{P}_3+d{P}_4=0$, where all $a, b, c, d$ are not equal to zero, then we say planes $P_1,P_2,P_3,P_4$ are linearly dependent planes).