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

There are $n$ straight lines in a plane, no $2$ of which are parallel & no $3$ pass through te same point Their points of intersection are joined. Show that the number of fresh lines introduced is $\frac{n\left(n-1\right)\left(n-2\right)\left(n-3\right)}{8}$.

There are $20$ books on Algebra & Calculus in our library. Prove that the greatest number of selections each of which consists of $5$ books on each topic is possible only when there are $10$ books on each topic in the library.

Find the number of ways to invite one of the three friends for dinner on $6$ successive nights such that no friend is invited more than $3$ times.

The straight lines $l_1,l_2$ and $l_3$ are parallel & lie in the same plane. A total of $m$ points are taken on the line $l_1,n$ points on $l_2$ and $k$ points on $l_3$. How many maximum number of triangles are there whose vertices are at these points?

In how many ways can you divide a pack of $52$ cards equally among $4$ players. In how many ways the cards can the divided in $4$ sets, $3$ of them having $17$ cards each & $4^{\mathrm{th}}$ with $1$ card?

(a) How many divisors are there of the number $21600$? Also find the sum of these divisors.

(b) In how many ways the number $7056$ can be resolved as a product of $2$ factors.

(c) Find the number of ways in which the number $300300$ can be split into $2$ factors which are relatively prime.