G Tewani Solutions for Chapter: Permutation and Combination, Exercise 3: DPP

There are two sets of parallel lines. Their equations are $x\cos \alpha +y\sin \alpha =p$ and $x\sin \alpha -y\cos \alpha =p$; $p=1, 2, 3\dots n$ and $\alpha \in (0, \frac{\mathrm{\pi }}{2}).$ If the number of rectangles formed by these two sets of lines is $225$, then the value of $n$ is equal to

The number of rectangles that can be obtained by joining four of the $12$ vertices of a $12$-sided regular polygon is

The interior angles of a regular polygon measure $150°$ each. The number of diagonals of the polygon is

Number of ways in which $7$ green bottles and $8$ blue bottles can be arranged in a row if exactly $1$ pair of green bottles is side by side is (assume all bottles to be alike except for the colour)

Which of the following is not the number of ways of selecting $n$ objects from $2n$ objects of which $n$ objects are identical?

A seven-digit number is in the form of $abcdefg$ ($g, f, e\dots$ etc.,, are digits at the units, tens, hundred place, and so on), where $a<b<c<d>e>f>g$ and $a, b, c, e, f, g$ are different digits. The number of such numbers is

The number of all six-digit numbers using digits $1, 2, 3, 4, 5$, such that any digit that appears in the number appears at least twice is

All the five-digit numbers $N=abcde$ having the property $a<b<c<d<e$ are arranged in the increasing order of their magnitude. The ${97}^{\mathrm{th}}$ number in the list does not contain the digit